Heat oven to 400°F. Grease a 9×13-inch baking dish.
Warm oil in a large skillet over medium-high heat. Brown the beef, breaking it up, 5–7 minutes. Drain excess fat.
Stir in onion and bell pepper. Cook until softened, about 4 minutes. Add garlic and cook 30 seconds.
Mix in tomato sauce, ketchup, Worcestershire, mustard, brown sugar, chili powder, paprika, 1 tsp salt, pepper, and broth. Simmer until thick, 5–7 minutes.
Spread the sloppy joe mixture into the baking dish. Sprinkle 1 cup cheddar over the top.
Whisk cornmeal, flour, sugar, baking powder, baking soda, and a pinch of salt in a bowl.
Add milk and melted butter. Stir just until combined. Fold in 1/2 cup cheddar.
Spoon batter evenly over the meat. Sprinkle remaining 1/2 cup cheddar on top.
Bake until the cornbread is golden and set in the center, 20–25 minutes. Rest 10 minutes before serving.