Preheat smoker to 275°F.
In a large bowl, combine ground chuck, ground round, Heinz 57 Sauce and A1 Steak Sauce.
In a medium saucepan, heat oil over medium-high heat. Add mushrooms and onion and cook until softened, about 25 minutes. Sprinkle lightly with garlic salt. Cut cheese into 16 squares.
Using a ⅓ cup measure, shape beef mixture to make 16 patties. On eight of the patties, place one square of cheese, then 1½ tablespoons of the mushroom mixture and top with another square of cheese. Place another plain patty on top and pinch edges to seal. Reshape the edges with your fingers to round out. Sprinkle lightly with pepper.
Place burgers on third rack of smoker and smoke for 1 hour or until internal temperature reaches 145°F and meat is no longer pink inside.
Place onto your favorite bun with all of your favorite toppings and enjoy!