Butter the brioche buns and toast them face-down in a skillet until golden brown. Set aside.

In a large skillet over medium-high heat, melt the butter. Add diced onions and bell peppers, cooking for 5-7 minutes until soft. Add minced garlic and cook 1 more minute.

Push veggies to the side and add ground beef. Break it apart and cook until thoroughly browned and no pink remains. Drain excess grease, leaving a little in the pan.

Sprinkle flour over the meat mixture and stir constantly for 1-2 minutes to cook the raw flour taste.

Pour in beef broth, Worcestershire sauce, ketchup, and Dijon mustard. Bring to a simmer, then reduce heat to low and cook for 3-5 minutes until sauce thickens and coats the meat. Season with salt and pepper.

Smooth the meat mixture into an even layer in the skillet. Turn off the heat.

Layer provolone cheese slices directly on top of the hot meat. Cover the skillet with a lid for 2-3 minutes until cheese is completely melted.

Scoop generous portions of the cheesy meat mixture onto the toasted brioche buns and serve immediately.