This is also called a patty melt: ground meat and cheese nestled between slices of bread (any kind will do) and grilled. Prewrapped slices of processed cheddar cheese are the only way to go, as they melt better. We do it here far lighter than you're likely to find it on the road.
Melt 1 teaspoon butter in a large nonstick skillet coated with cooking spray over medium-high heat. Add onion, and cook for 4 minutes or until golden brown, stirring frequently. Reduce heat; cook 10 minutes or until tender, stirring occasionally. Set aside.
Combine beef, pepper, and salt in a medium bowl. Divide the beef mixture into 4 equal portions, shaping each into a 1/4-inch-thick oval patty. Place the patties on a broiler pan coated with cooking spray; broil 4 minutes on each side or until done.
Place 1 cheese slice over each of 4 bread slices; top each slice with 1 patty and 3 tablespoons onion mixture. Cover with the remaining bread slices. Melt 1/2 teaspoon butter in pan coated with cooking spray over medium heat; add 2 sandwiches to pan. Cook for 4 minutes on each side until browned and cheese melts. Repeat the procedure with 1/2 teaspoon butter and the remaining sandwiches.
Very quick recipe. I made it exactly how it describes, but we used white bread. Which unfortunately my husband & I were very disappointed with. We were really looking forward to trying it, but it tasted like a normal grilled cheese sandwich with ground beef added to it. So, hopefully it will be more flavorful with actual sourdough bread.