Make two burger patties from the ground beef, then season with salt + pepper just before cooking.
Heat a large cast-iron skillet on high until it begins to smoke. Drizzle the skillet with just enough canola oil to coat the bottom with a thin layer.
Add the burgers and green onions to the skillet and reduce heat to medium-high. Cook for about 4 minutes on the first side, or until they are nicely browned. Once you have flipped the burgers, add the brie cheese and caramelized onions on top of the cooked side to start the melting process. Cook for 3 to 4 minutes on the second side, or until they are to your desired doneness and the cheese is melted.
Remove burgers from skillet to rest. Toast the bread on each side in the hot skillet.
Assemble with the bread, mayonnaise, burger, caramelized onions and top with the cooked green onions.