In a small bowl, whisk together mayonnaise, barbecue sauce, 1 tablespoon bourbon, ¼ teaspoon salt, and ¼ teaspoon pepper. Refrigerate until ready to use.
In a large bowl, stir together pork, bread crumbs, garlic, Worcestershire, egg, remaining ¼ cup bourbon, remaining ½ teaspoon salt, and remaining ½ teaspoon pepper. Divide pork mixture into 4 portions, and shape each portion into a 5-inch patty.
Heat a large cast-iron skillet over medium heat. Add patties to skillet; cook for 5 minutes per side. Add cheese; cook until cheese is melted, about 1 minute more. Serve patties on rolls with lettuce, onion, and bourbon barbecue sauce.
Recipe by Southern Cast Iron at https://southerncastiron.com/bourbon-barbecue-burgers/