Mash the corned beef into a saucepan, and stir in ketchup, mustard, Worcestershire sauce, onion, and white Cheddar cheese over medium heat. Bring to a boil, reduce heat to medium-low, and simmer until the cheese has melted and the onion is tender and translucent, about 10 minutes. Fill each hot dog bun with about 1/2 cup of the mixture, and wrap each filled bun in aluminum foil. Place the wrapped buns onto a baking sheet.
Bake in the preheated oven until buns are slightly toasted, about 15 minutes. If desired, make a big batch of filled and wrapped buns, freeze them, and bake frozen for 35 minutes.