While we aren't known for our hamburgers, LA has some pretty good ones!
Judice Inn in Lafayette, LA is my favorite. Hand formed patty grilled to crusty & juicy perfection, dressed (lettuce, etc...) , with their special sauce (ketchup based). Only chips and soft drinks (or beer) as sides. Wonderful!
Pete's burgers (Lafayette, LA) are also good (Big Pete is on special most of the time).
Finally, Morvant's bar in Youngsville, LA serves a GREAT burger. Grilled patty and simple presentation.
While many people will come to LA with Cajun food on their mind, hamburgers are a small part of our community.