Cook the bacon in a 5-quart Dutch oven until crispy, remove to paper towels to drain. Add the ground beef to the pot with the bacon grease and cook until beginning to brown, breaking the meat into small pieces. Add the chopped onion and garlic and cook until the onions are soft and the meat is cooked through. Drain the fat from the pot, then return it to the heat. Stir in the Worcestershire sauce, then add the chicken broth and bring to the boil. Reduce the heat to simmer.
Meanwhile, in a small bowl, whisk together the milk and the flour until smooth. While the soup is simmering, stir in the flour and milk and stir and simmer until the soup is slightly thickened. Add the grated cheese a handful at a time, stirring to melt after each addition. When all the cheese is incorporated, stir in the mustard and the ketchup and add salt and pepper to taste. Crumble the cooked bacon and stir into the soup. Cover and simmer the soup for 10 minutes.
While the soup is simmering, spread each side of the hamburger buns with butter, cut into thin strips and toast in a toaster oven or under the broiler until lightly toasty. Serve the soup immediately, accompanied by the bun croutons. If you'd like, you could top with extra crumbled bacon, pickle relish, more grated cheese-anything you like on your cheeseburger!