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 (200º to 260ºC).
Mix the patty ingredients, and then gently form four patties of equal size, each about 18 millimeters thick. Make a hole in the center of each patty for the cheese.
Trim away the rind of the Brie and cut into four equal portions. Bury one cheese portion into the hole of each patty and cover with about 8 millimeters 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.
Brush the cooking grates clean. Grill the patties over 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.
