The two separate calls, which lasted about 10 minutes and 12 minutes respectively, occurred in May shortly after Nikol Pashinyan was sworn in as Armenia's new prime minister.
On the other end of the line, however, were Alexey Stolyarov and Vladimir Kuznetsov.
What did Juncker and Mogherini say?
Conversation with Juncker
The call with Juncker appears to have been taken on May 9, as he refers to US President Donald Trump's decision to abandon the Iran nuclear the day before — a move the EU leader called a "mistake" on the phone.
"This is a major concern for us and for me personally because I do think that this is a major mistake, the one that Trump had made yesterday night," said Juncker.
"We as Europeans, we are sticking to the Iran deal. France, Germany, Britain, others, including the European Union itself, we will stick to that agreement and we don't want to destabilise the region. And as Iran is your neighbour, you have the highest interest in having the whole region being stabilised so we are working in that direction."
Trump was referred to again when Stolyarov, pretending to be Pashinyan, said Armenia was receiving pressure from the US and asked for the EU to mediate. Juncker responded: "We have to explain again and again the world to Mr Trump, he doesn’t understand what it’s about. We will make sure that he has a better understanding of what is happening in Armenia."
When Stolyarov said his political party had experienced "resistance from Russia", Juncker urged him to go forward and maintain good ties with Putin's government.
"And with Russia, we are not asking to break your relations with Russia. It’s important that you will have a meeting with Vladimir Putin, I think on the 14th of May or something like that, that's good. It's good. He's a personal friend of mine so I can talk with him whenever it's needed," Juncker said.
Read more | Full transcripts of the prank call on Juncker, Mogherini
The two also talked about Juncker's love for Armenian music, and Stolyarov said a concert was being played by rock band System of a Down, whose lead singer is Armenian-American, as the band's recorded track is heard in the background.
Stolyarov ended the conversation by saying that "team Kardashian" would like to work in the ministry of foreign affairs.
"Oh yeah, that's good," said Juncker.
Conversation with Mogherini When it came to Russia, Mogherini — like Juncker — wasn't baited into squabbling about the Russian Federation.
When Stolyarov referred to the geopolitical pressure of neighbouring countries like Turkey and Azerbaijan, as well as from Russia, Mogherini said Armenia can count on the EU.
"You’re not living in an easy environment but you can count on us for sure for having a friend on which you can count. It’s true it’s a challenging region and again you’re facing different sorts of pressures and challenges but I'm sure that there could be a way to smooth the tensions and try to find cooperative approaches."
"Count on us anyway every time for discussing and finding a way forward and cooperating very, very closely."
'Improving our procedures'
Spokespeople for both Juncker and Mogherini did not question the authenticity of the tapes when contacted by Euronews.
Juncker's spokesperson said: "The President loves Armenia. Whoever calls him, even on a European public holiday, he is happy to discuss its music and cuisine."
A spokesperson for Mogherini said in a written statement: "We have regular contacts with Armenian authorities. HRVP Mogherini most recently met (the) PM of Armenia Pashinyan last week in the margins of NATO summit."
"When it comes to how calls are arranged, we have clear procedures in place, and we are constantly improving them."
The Kremlin or pranksters?
Critics, mainly from the West, have long accused the two pranksters — Stolyarov and Kuznetsov, who are more well known as their stage names Lexus and Vovan — of having ties to the Kremlin due to their seemingly frequent ability to get leaders in Europe on the other end of the line, including former Foreign Secretary Boris Johnson.
Stolyarov denies the claim.
When asked if he supports Putin, Stolyarov tells Euronews: "I support many things in foreign politics that [Putin's] team did. With some things in our internal politics I disagree. But, anyway, he is elected Russian president."
"We do our job without orders from the government. We do that as private persons and it’s not (the) position of Kremlin despite some government workers like it. Some people from opposition also like it."
Stolyarov also points to how they have carried out similar stunts on Russia's own government. "We pranked already our governors, some Russian ministers, deputies etc. It was earlier. Sometimes we make such calls, too. But geopolitics is more interesting."
In the case of the call with Juncker and Mogherini, one big concern is a security breach.
Anthony Glees, the professor of politics at the University of Buckingham who directs its Centre for Security and Intelligence Studies, said these stunts can threaten national security.
"I'd say that pranksters can represent a real threat to national security and good governance even if their purpose is simply to give the public a laugh."
"This is because a head of state, a senior government minister — someone like Boris Johnson when he was Foreign Secretary — will not only be part of the inner circle of policy-makers but privy to the UK's national security secrets and those of other nations."
"What might seem like a prank could have huge implications if secret policy strategies e.g. over Brexit were being explored and therefore 'prank' is really the wrong word."
"Unwitting espionage disguised as a prank sums it up."
Glees also pointed out that the Lisbon Treaty gives the EU "no responsibilities in respect of the national security of member states so there would not be EU secrets in this respect".
He recommended simple practices that all too often go overlooked, such as frequently changing mobile numbers and having MPs hand their phones over during high-level talks.
"We know that [German Chancellor] Merkel's private mobile phone number had been easily accessed by the NSA because she had not changed it in years, despite security advice to do so."
"Ministers also use their mobiles recklessly — hence, [UK Prime Minister] May's order that at her Chequers meeting (with Trump) last week, all mobiles had to be handed in."
The duo has done it time and time again. From Elton John to Turkish President Recep Tayyip Erdogan, both Stolyarov and Kuznetsov have orchestrated pranks onto the front pages. Stolyarov insists that they're not doing it to be malicious and the targets of their pranks are not politically motivated. The purpose of the prank, however, isn't solely to have a laugh.
Stolyarov says it's about accessing information — and also having a laugh.
"Sometimes it's to get information — very important information — that's one of our goals. The second is entertainment." It's a genre that Stolyarov describes as "prank journalism".
"It's not just a joke, it's an element of journalism."
But the duo draw a line when it comes to politicians' private lives.
"I would not release their private secrets, but if their secret is about society or about their populations, then, of course, it’s OK to publish it," says Stolyarov.
He also praised Juncker and Mogherini's "kindness" and sense of humour during the call.
For Glees, the risks associated with these pranks far outweigh the reward of a laugh.
"In the Second World War there were posters all over London saying 'careless talk costs lives'. It's still so today and in truth the pranksters aren't funny, they're villains," he said.
Alice Cuddy and Anastassia Gliadkovskaya contributed to this report.