In a sauté pan over medium heat, combine onions with butter and oil and sauté until onions are dark golden brown and caramelized. Set aside to cool.
Prepare a medium-low fire on the grill.
To assemble quesadillas, divide onions among tortillas, spreading them over one half of the tortilla. Next, lay Brie over onions, followed by the cooked lobster meat, and fold over to form half circles.
Brush the top and bottom of each tortilla lightly with olive oil.
Carefully place quesadillas on the grill for 2-3 minutes. Using a spatula, flip and cook for another 2-3 minutes, or until the tortillas are toasted and cheese is melted.
Transfer quesadillas to a cutting board and, using a well-sharpened knife, slice into 4 wedges. Serve with salsa.
