Preheat the oven to 425°F. Spread the frozen tater tots on a sheet pan and bake for 20-25 minutes, until thawed, warmed, and just starting to brown. They won't be fully crisp yet.
While the tots are baking, heat the oil in a large skillet over medium heat. Add the ground beef and cook until browned. Season with the salt and pepper.
Add the onion and bell pepper. Cook 4-5 minutes until softened. Stir in the garlic and cook 30 seconds.
Add the ketchup, water, tomato paste, mustard, brown sugar, and Worcestershire sauce. Bring to a simmer and cook 5-10 minutes, stirring occasionally, until the liquid thickens slightly but stays saucy. Taste and add more salt or mustard if needed.
Transfer the sloppy joe filling to a 13x9 glass or ceramic baking dish (I don’t bother to grease it).
Carefully arrange the hot tots on top of the filling (if you used a 32 ounce bag, they won’t all fit) and sprinkle with the cheddar cheese.
Bake at 425°F for 15-20 minutes, until the tots are crisp on top and the filling is bubbling around the edges. Tip: If the casserole is bubbling but the cheese has not browned, broil for 60-90 seconds.
Remove the casserole from the oven and let cool slightly before serving.