I really liked the Cajun Shrimp Burger at Mason's Grill off Old Jefferson....
More often than not the best burger will be found in a bar that serves food(bar food)like burgers.
For my tastes... In Baton Rouge, Riverside Patty off Siegen Lane or the original one Downtown.
The BIG Pete at Pete's on Johnston in LAF with the grilled onions is a FREAKING beast !!!!!!!!!!