Finely chop 3 slices thick-cut bacon, if using. Place in a medium skillet over medium heat and cook until the fat begins to turn semi-translucent, about 4 minutes. Transfer the bacon to a paper towel-lined plate.
Place the bacon, 1 1/2 pounds ground beef, 2 teaspoons smoked paprika, and 1 tablespoon plus 1 1/2 teaspoons Worcestershire sauce in a large bowl. Using your hands, quickly and gently mix until just combined.
Divide the mixture into 4 patties (about 6 ounces each). Season both sides generously with kosher salt and freshly ground black pepper.
Heat an air fryer to 380°F. Working in two batches if needed, place the patties in a single layer in the basket so that they are not touching. Air fry for 4 minutes. Flip the patties and air fry to desired doneness, 5 to 7 minutes more for medium. Top each patty with a slice of cheese, if using, during last minute of cooking. Serve in buns with toppings if desired.