In a large skillet over medium heat, heat vegetable oil. Add onion and bell peppers and cook, stirring often, until softened, about 5 minutes.

Stir in the garlic and cook until fragrant, about 30 seconds, then add the beef. Cook until no longer pink, 6 to 8 minutes. Drain fat

Stir in beef broth, ketchup, brown sugar, and Worcestershire and season generously with salt and pepper. Simmer until most the liquid has reduced and the mixture is thick, 5 minutes.

Top beef with provolone and cover skillet with a tight-fitting lid to melt, about 2 minutes.

Remove from heat and serve cheesy ground beef mixture in hamburger buns. Serve warm.