To caramelize the onions, heat olive oil in a large saute pan over medium low heat. Once hot, add onions and sprinkle with salt.
Cook slowly, stirring frequently, until a caramel brown colour and soft, about 15 to 20 minutes. Do not let them get crispy. Reduce heat if necessary to keep them from burning or crisping.
Keep saute pan for cooking mushrooms.
For the burgers, combine beef, salt and pepper in a large bowl. Mix by hand to combine well and divide into 6 even portions.
Take 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.
Cook 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. Meat should register at least 125°F (50°C) to 130°F (55°C) on an instant-read thermometer. Remove from grill and let sit 5 minutes.
Cook mushrooms while burgers are grilling. Add butter to saute pan and melt over medium heat. Add mushrooms and a little salt and saute until mushrooms are browned on both sides and tender.