Preheat the oven to 350 degrees F (175 degrees C). Lightly grease an 11x7-inch baking dish.
Heat a large skillet over medium-high heat. Cook and stir ground beef and onion in the hot skillet until meat is browned and crumbly, 5 to 7 minutes. Drain and discard grease. Stir in ketchup, 2 teaspoons Dijon mustard, 2 teaspoons Worcestershire, and garlic powder. Season with salt and pepper to taste.
1 pound ground beef, ½ onion, chopped, 2 tablespoons ketchup, 3 teaspoons Dijon mustard, divided, 3 teaspoons Worcestershire sauce, divided, ½ teaspoon garlic powder, salt and ground black pepper to taste
Cut rolls in half horizontally without separating each roll from one another. Place bottom halves in the prepared baking dish. Spread meat mixture on rolls, top with Monterey Jack cheese, and cover with roll tops.
1 (8 ounce) package shredded Monterey Jack cheese
Combine butter, brown sugar, remaining Dijon mustard, and remaining Worcestershire in a microwaveable bowl; cover loosely and microwave until butter is melted, about 1 minute. Mix and pour mixture over the top of the rolls.
3 teaspoons Dijon mustard, divided, ¼ cup butter, 1 tablespoon brown sugar, 3 teaspoons Worcestershire sauce, divided
Bake, covered, in the preheated oven for 15 minutes. Uncover and bake until cheese is melted and rolls are golden brown, about 10 minutes more.