Thinly slice the shallot.

Heat olive oil in a medium skillet over low heat.

Add shallots to the skillet.

Cook until browned but not scorched, about 20 minutes, stirring often.

Let cool to room temperature.

Prepare the grill for direct cooking over high heat.

In a large bowl, gently mix ground chuck, breadcrumbs, beef broth, kosher salt, and black pepper.

Shape the mixture into 4 patties of equal size and thickness.

Make a hole in the center of each patty.

Trim away the rind of the Brie cheese.

Cut the brie cheese into 4 equal portions.

Bury the brie, about 1/4 ounce for each patty, into the hole of each patty.

Close the opening to seal the cheese inside, ensuring at least 1/3 inch of meat covers the cheese.

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.