In a small bowl, combine beef, ketchup, mustard and salt, mixing lightly but thoroughly. Cut onion into 1/2-in. slices; separate into rings. Fill 8 slices with half of the beef mixture (save remaining onion rings for another use). Top each with a piece of cheese and remaining beef mixture.
In a shallow bowl, mix flour and garlic powder. Place bread crumbs and eggs in separate shallow bowls. Dip filled onion rings in flour to coat both sides; shake off excess. Dip in egg, then in bread crumbs, patting to help coating adhere.
In an electric skillet or deep fryer, heat oil to 350°. Fry onion rings, a few at a time, until golden brown and a thermometer inserted into beef reads 160°, 2-3 minutes on each side. Drain on paper towels. If desired, serve with spicy ketchup.