The body that nominates the Commission isn't elected.
In theory the Commission is mostly made up of civil servants who answer to commissioners, who are themselves nominated by each country's own government or civil service. Each commissioner has one area of responsibility only, and they answer to the head of the Commission who is their boss. So someone in the UK votes for a politician, who votes for a party leader, who appoints some ministers, and those ministers may or may not have much of a say in whoever gets nominated to be a commissioner - one of many. But there is at least a path there, even if long and indirect and the person your vote ends up influencing doesn't do anything important to your country or needs.
In practice it doesn't actually work that way. In practice, the head of the Commission has veto power over the nominations. They aren't supposed to according to the treaties but the treaties are ignored. This means that in reality it's the head of the Commission who picks the Commissioners, because they can just reject anyone who isn't sufficiently aligned with their own agenda.
So that leaves the question of how the head of the Commission is picked. Once again there is theory and practice. In theory, it's a decision of the heads of each state that they take together to select some candidates, and the Parliament then gets to vote for their preferred candidate. In practice ... nobody knows how the head is picked. Ursula von der Leyen was recently re-appointed despite being plagued by scandals and having a long career of failing upwards. Parliament was sidelined by giving them a voting list with only one candidate on it (her). Seek out an explanation of how she got this job and you won't find one because:
1. The heads of state don't talk about how they decide as a group. Is it a vote? Some sort of horse trading? Do they take it in turns? Are they even all able to take part? Nobody knows.
2. There's no record of which country voted for who, or why.
3. The process by which someone even becomes a candidate is unclear.
4. Because no head of state has any control over who gets onto the candidate list, they never talk on the campaign trail about how they will "vote" (assuming that's how it works) for who runs the EU.
In other words, the process is entirely secret. The potential for corruption is unlimited.
So when critics say the EU Commission is a bunch of unelected bureaucrats, they are right and those who argue otherwise here on HN are wrong. People who got their jobs via a process so opaque and indirect that how it functions can't be explained, not even in principle, cannot claim to be democratically selected.