Beef & Cheddar Sandwich

Ingredients

• 2 tablespoons butter
• 2 tablespoons flour
• 3/4 cup milk
• 2 cups cheddar cheese, freshly shredded
• salt, to taste
• pepper, to taste
• 1/2 cup ketchup
• 1 tablespoon apple cider vinegar
• 1 tablespoon brown sugar
• 2 tablespoons water
• 1/4 teaspoon garlic powder
• 1/2 teaspoon onion powder
• hot sauce, optional, to taste
• 8 onion hamburger buns, split
• 1 pound thinly sliced deli roast beef, rare
• beef broth, optional, to taste