Add the bourbon to ¼ of a cup of the barbecue sauce and simmer the mixture in a small saucepan over medium heat until slightly thickened, about 2 minutes. Set aside.
Combine the ground beef, the remaining ¼ cup of barbecue sauce, salt, and pepper in a large bowl. Mix just until combined and then gently shape into 4 patties.
Grill the patties over medium heat, about 5 minutes per side, or until a thermometer inserted into the center of the patty reads 160 degrees F.
Top each patty with a slice of cheese and grill one minute longer.
Place the patties on the toasted buns and layer with lettuce, tomato, red onion slices, pickles, and bourbon barbecue sauce, to taste.