Place onion and celery into a pot and cover with water; bring to a boil. Simmer until tender, about 3 minutes. Drain.
Heat a large skillet over medium-high heat; cook and stir beef until it is crumbly, evenly browned, and no longer pink, about 5 minutes. Drain and discard any excess grease. Stir in onion mixture, ketchup, tomato soup, 1/2 can tomato paste, garlic salt, salt, and pepper; set aside to cool.
Mix remaining 1 1/2 cans tomato paste, oregano, and sugar in a small bowl. Season with salt and pepper.
Butter each English muffin half, then spread with tomato paste mixture. Divide ground beef mixture over each muffin half and top with strips of American cheese. Place open-faced sandwiches on large baking sheets.
Bake in the preheated oven until cheese melts, about 15 minutes.