Heat a large nonstick skillet over medium heat. Coat pan with cooking spray. Add shallots and garlic to pan; cook 1 minute or until tender, stirring frequently. Increase heat to medium-high. Add mushrooms to pan; cook 10 minutes or until moisture evaporates, stirring occasionally. Combine broth, steak sauce, and cornstarch, stirring with a whisk. Add broth mixture to pan; bring to a boil. Cook 1 minute or until thickened, stirring constantly. Stir in 1/4 teaspoon pepper. Remove mushroom mixture from pan; cover and keep warm. Wipe pan with paper towels.
Combine remaining 1/4 teaspoon pepper, ketchup, and Worcestershire sauce in a large bowl, stirring with a whisk. Add beef to bowl; toss gently to combine. Shape beef mixture into 4 (1/2-inch-thick) patties; sprinkle evenly with salt.
Heat pan over medium-high heat. Coat pan with cooking spray. Add patties to pan; cook 4 minutes. Turn and cook 3 minutes or until desired degree of doneness. Cover each patty with a slice of cheese, melt slightly. Place 1 lettuce leaf and 1 tomato slice on the bottom half of each roll. Top each serving with 1 patty, about 1/4 cup mushroom mixture, and top half of roll.