Brown the ground beef, onion, and celery together in a large skillet over medium heat until the beef is no longer pink and the vegetables begin to soften. Drain off excess fat thoroughly so the make-ahead filling will not become greasy or loosen the buns.

Stir in the tomato sauce, ketchup, prepared yellow mustard, brown sugar, salt, and black pepper. Bring the mixture to a gentle simmer and cook uncovered for about 10 minutes, stirring occasionally, until thick enough to sit in the buns.

Split the hamburger buns, keeping the tops and bottoms together as pairs. Spoon the warm beef mixture onto the eight bun bottoms, place one American cheese slice over each portion, and replace the bun tops.

Wrap each assembled bunwich securely in aluminum foil, making sure the seams are closed. Refrigerate the sandwiches for several hours or overnight until you are ready to bake them.

Heat the oven to 350°F (175°C). Arrange the foil-wrapped bunwiches on a baking sheet and bake for 20–25 minutes, until the sandwiches are hot throughout and the cheese has melted.

Open the foil carefully because steam may escape, then serve the bunwiches immediately. Keep the sandwiches in their foil for serving if you want them to stay warm during a busy family meal.