For the burgers: Divide the meat into four equal potions. Form each portion loosely into a disk shape, making a deep impression in the center with your thumb. Season both sides of each burger to taste with salt and pepper.

Prepare and preheat a grill, grill pan, or cast iron griddle. Cook the burgers until slightly charred and cooked through, about 5 minutes per side. If using very fresh and high quality ground beef, feel free to cook to medium or medium-rare, about 3 to 4 minutes per side. Remove the burgers from the heat and add the brie cheese. Tent with aluminum foil and allow the burgers to rest for 5 minutes, or until the juices are redistributed and the cheese is melted.

For the garlic mayonnaise: Combine all the ingredients in a small bowl and refrigerate until ready to use.

To finish: Top each burger with arugula and caramelized onions. Spread garlic mayo on each burger bun and sandwich the burgers. Serve immediately.