In a large non-stick or cast-iron skillet over medium heat, add chopped bacon. Cook for 5-6 minutes until bacon is crisp. Remove bacon from skillet and drain on paper towels (reserve for later). Drain some of the bacon grease but leave at least 2 tablespoons in the skillet.
Next, add onions to the bacon grease. Cook over medium heat until onions are softened (about 5 minutes). Then, reduce heat to LOW and cook onions for another 35-40 minutes until onions are soft and caramelized. They should be a rich brown color.
Stir in brown sugar, reserved bacon, balsamic vinegar, and salt. Cook 2-3 minutes more until all flavors are incorporated.
Transfer jam to a bowl. Serve with burgers at room temperature. If you want to make it ahead, jam will keep in an airtight container for 4-5 days in the fridge.
Form meat into (4) 4-inch patties (about ½ inch thick). Season both sides with sea salt and pepper.
Cook burgers in either a skillet or on the grill until internal temperature reaches 155-160 degrees, flipping one time (about 2-3 minutes on each side).
After flipping the burger, add 4 slices of brie to each burger and cover to ensure the cheese melts completely. Let burgers rest, covered for a few minutes--this will allow the juice to redistribute and the cheese to get extra melty!
To assemble burgers, spread the bacon onion jam on the bottom of the bun. Top with burger, more jam and arugula. Serve immediately.