You can usually see if someone is registered to vote if you have their name, location, and some other details. But as far as I can tell, finding out if they actually cast a vote in a particular election requires a request submitted to the county clerk where they're registered
Though I saw your other comment on another thread and I wouldn't be surprised if he did vote in both elections. I am surprised someone could verify it !
One thing that really captures the problem is that you were able to quickly fact check this and so could the NYT, but you as a regular citizen actually did and the NYT as an incredibly well-resourced team of, ostensibly, professional fact-finders did not.