Season the beef patties with salt and pepper on both sides. Preheat a skillet over medium-high heat and cook the patties for 3-4 minutes per side, until a golden-brown crust forms.
When the patties are nearly cooked through, top each with a slice of brie cheese. Cover the skillet and allow the cheese to melt.
To prepare the caramelized onions, melt the butter in a skillet over low heat. Add thinly sliced onions and cook, stirring occasionally, for 10 minutes, or until softened and golden brown.
Toast the hamburger buns in a skillet or toaster until the insides are golden brown.
Wash the arugula and slice the tomatoes into thin rounds.
Assemble the burgers: spread French mustard on the bottom bun, top with arugula, the brie-topped beef patty, caramelized onions, and tomato slices.
Top with the remaining bun and gently press down. Serve immediately with fries or a green salad.
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