To caramelize the onions, heat the oil in a large saute pan over medium low heat. Once hot, add the onions and sprinkle with salt.
Cook slowly, stirring frequently, until they become tender and a deep amber, about 15 to 20 minutes. Do not let them get crispy. Reduce the heat if necessary to keep them from burning or crisping.
Combine the beef, salt, and pepper in a large bowl. Mix by hand to combine well and divide into 6 even portions.
Take a little more than half of each portion and form into a thin patty. Top each thin patty with some of the brie and onions, then top with remaining half of each portion and form around cheese and onions, into one large patty. Seal the edges well.
Cook the burgers on pre-heated grill about 5 to 6 minutes per side for medium. Cook slightly less for medium-rare and slightly more for medium-well. (Note: The USDA says burgers should be cooked to 160F for food safety. This is well done. I prefer mine medium rare but I must give you the proper safety information!)
Serve with lettuce, sliced tomatoes, and pickles, if desired.
You can also pan fry, oven bake or air fry these burgers.