In a saucepan, combine soy sauce, pineapple juice, brown sugar, honey, garlic, and ginger.
Bring to a simmer, then whisk in cornstarch slurry until thickened. Set aside.
In a bowl, mix ground beef, egg, breadcrumbs, salt, and pepper. Form into 4 patties.
Grill patties over medium heat for 4–5 minutes per side, until cooked through.
In the last minute of grilling, add cheese slices on top of patties.
Grill pineapple rings for 2 minutes per side until caramelized.
Toast burger buns lightly.
Assemble burgers with lettuce, tomato, patty, grilled pineapple, and teriyaki sauce.
Serve hot with extra teriyaki sauce on the side.