Prepare a charcoal or gas grill for medium heat (350° to 450°; you can hold your hand 5 inches above cooking grate only 5 to 7 seconds). In a small bowl, combine pineapple, barbecue sauce, and soy sauce. Set aside.
In a medium bowl, combine pork, bread crumbs, onion, ginger, brown sugar, cayenne, and salt. Shape mixture into 4 patties. Cook, turning once, until slightly charred and cooked the way you like them, about 7 minutes total for medium.
Put each burger on a muffin half. Top each with 2 basil leaves, about 2 tablespoons sauce, and the muffin top.