In a saucepan over medium heat, combine soy sauce, brown sugar, honey, grated ginger, and minced garlic. Stir and simmer until sugar dissolves.
Add the cornstarch slurry and cook, stirring, until the sauce thickens, about 2–3 minutes. Remove from heat and let it cool.
Season the ground beef with salt and black pepper. Form into 4 equal patties with a shallow indent in the center.
Preheat the grill or grill pan to medium-high heat. Grill pineapple rings for 2 minutes per side until caramelized. Set aside.
Grill the beef patties for 4–5 minutes per side, or until desired doneness. Add a slice of cheddar cheese during the last minute to melt.
Grill turkey ham slices briefly just to warm through.
Assemble each burger with a toasted bun bottom, lettuce, teriyaki-glazed beef patty with melted cheese, turkey ham, grilled pineapple, and optional red onion.
Drizzle additional teriyaki sauce over the toppings and spread mayonnaise on the top bun if desired. Close the burger and serve.