In a medium skillet over low heat, 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 high heat (450° to 550°F).
In a large bowl gently mix the patty ingredients and shape into 4 patties of equal size and thickness. Make a hole in the center of each patty for the cheese.
Trim away the rind of the Brie and bury the cheese, about ¼ ounce for each patty, into the hole of each patty. Close the opening to seal the cheese inside. It’s important that there is one-third of an inch of meat on the top and bottom of the cheese so it doesn’t seep out.
Brush the cooking grates clean. Grill the patties over direct high heat, with the lid closed as much as possible, until cooked to medium, 8 to 10 minutes, turning once when the patties release easily from the grate without sticking. During the last minute of cooking time, toast the buns, cut sides down, over direct heat.
Assemble the burgers with shallots, mustard, and arugula. Serve warm.