Preheat oven to 400°F and grease a 9x13-inch baking dish.
In a large skillet over medium-high heat, brown ground beef until well-caramelized, about 10 minutes, then add onion, bell pepper, and garlic and cook until softened, about 5 minutes.
Stir in ketchup, brown sugar, Worcestershire sauce, mustard, salt, and pepper, then simmer for 5 minutes until mixture thickens slightly.
Pour the sloppy joe mixture into the prepared baking dish and spread evenly.
In a medium bowl, combine biscuit baking mix, milk, and melted butter, stirring just until combined.
Drop spoonfuls of biscuit dough over the meat mixture, then sprinkle shredded cheese over the top.
Bake for 20-25 minutes until biscuits are golden brown and cooked through, then let stand 5 minutes before serving.