I'm surprised everyone is saying 'what' is the best option here.. In my opinion, which works best. "I want to figure out which IS the best burger in the United States" or, I'd say.. "I want to figure out which burger is the best in the United States"

You'll actually want to say, "I want to figure out what the best burger in the United States IS." In English, the Subject comes before the Verb in most cases except questions. You don't need to move the verb in the second clause ("What the best burger in the United States is") because this isn't a question.

I agree with Mike on this but I wouldn't use the phrase "figure out" as it implies calculating. I would say "I want to know which burger is the best in the United States." implies experience. If I wanted to know which is the cheapest burger I would say. "I want to figure out the cheapest burger in the United States." So many ways to communicate an idea.

Melissa, Amanda, and Jessica are correct: "I want to figure out what the best burger IS in the United States" is correct, but, "I want to figure out what the best burger in the United States is," sounds better. They both have the subject first (burger) followed by the verb (is), but including the qualifiers with the subject (best burger in the United States) sounds better than injecting the verb in the middle of the description (best burger is in the United States).

Second option because it is an indirect question. For example: I want to know what she writes. Direct queston: what does she write?