In a medium skillet over low heat on the stove, combine the shallots with the oil and cook until they are browned but not scorched, about 20 minutes, stirring often. Let cool to room temperature.
Prepare the grill for direct cooking over medium–high heat (200ºC to 260ºC).
Mix the patty ingredients and then gently form four patties of equal size, each about 2 centimetres thick. Make a hole in the centre of each patty for the cheese.
Trim away the rind of the Brie and cut into four equal portions. Insert one cheese portion into the hole of each patty and cover with about 8 millimetres of the meat to seal the cheese inside. It’s important that the cheese is well covered so it doesn’t seep out.
Brush the cooking grates clean. Grill the patties over direct medium–high heat, with the lid closed, until cooked to medium doneness, 9 to 11 minutes, turning once. During the last 30 seconds to 1 minute of grilling time, toast the buns, cut sides down, over direct heat.
Build each burger on a bun with a patty, shallots, mustard and arugula. Serve warm.