Preheat the grill to medium and brush the grill grate with a high-heat oil such as avocado oil.
Combine the egg, carrot, garlic, 2 tablespoons of the teriyaki sauce, and onion powder into a bowl. Add turkey and mix until combined and shape into 4 patties.
Coat both sides of patties with nonstick spray, then grill, covered, 6-8 minutes per side. (A thermometer should measure and internal temperature of 165 degrees) Baste patties with the teriyaki sauce during the last 5 minutes of cooking.
Grill the pineapple slices until grill marks appear, about 3-4 minutes per side.
Serve on a bun or with lettuce and extra teriyaki sauce, if desired.