Brown the ground beef in a large skillet over medium heat with the chopped onion, celery, and green bell pepper. Cook for about 6 to 8 minutes, breaking the beef into crumbles, until it is no longer pink and the vegetables have softened.

Drain off the excess fat from the skillet before adding the sauce ingredients. Returning the drained beef and vegetables to the pan keeps the finished filling rich and flavorful without making it greasy.

Stir in the tomato sauce, ketchup, water, brown sugar, Worcestershire sauce, salt, and black pepper. Mix thoroughly so the sugar and seasonings are distributed throughout the beef mixture.

Bring the mixture to a gentle boil over medium heat, then reduce the heat to low. Keep the skillet uncovered and simmer for 15 to 20 minutes, stirring occasionally, until the sauce thickens.

Spoon the hot filling onto the six split hamburger buns and serve immediately. Lightly toasted buns are especially good because they provide a little crispness and hold up better beneath the saucy beef.