Did you mean ハンバーガー (hamburger in a bun) or ハンバーグ (just the meat)? You should be careful that the former is a loanword from 'hamburger', while the latter is a loanword from just 'Hamburg', so if you have a グ at the end, it shouldn't be a long vowel.
I think ２枚 is normally the counter for ハンバーグ, especially when you're buying it. I found this image on Rakuten. You can see the counter at the bottom.
I'm not sure if that would be the right counter for ハンバーガー, but I think in that case it would be normal just to use the generic つ counter.
Got it. In the context of ordering I think it's more usual to use the generic counter つ, although 個 would also work, especially if you were ordering only hamburgers, and nothing else.
If you're not ordering, but talking about ハンバーガー in some other context, then 個 would be the proper counter. In this case we use 個 simply because a hamburger is round and 個 is 'properly' for counting round things (but see the other reason to use 個 below). An example I found with Google:- マックのビッグマックはハンバーガー3個と値段がほぼ同じですが、あなたはどちらが得だと思いますか？ Makku no biggu makku wa hanbāgā sanko to nedan ga hobo onaji desu ga, anata wa dochira ga toku da to omoimasu ka? "A Mcdonald's Big Mac is roughly the same price as three hamburgers, but which do you think is better?"
However, the situation is confused a little bit by the fact that some people just use 個 as a generic counter instead of つ anyway. There's a whole explanation here ( http://www.imabi.net/countersiikovstsu.htm ), but basically some people think the 個 counter, which came from Chinese, sounds more 'sophisticated' than つ, which is natively Japanese. Conversely, you could use つ instead of the 'proper' counter 個 in the above example, but it would sound less sophisticated.