In a large bowl, gently combine the ground beef with 2 tablespoons of teriyaki sauce, salt, and black pepper. Mix just until combined and shape into four equal patties.

Add 1 tablespoon of vegetable oil to a large skillet or grill pan and heat over medium-high temperature.

Place the patties onto the heated surface and cook for 4 to 6 minutes per side, depending on your preferred doneness. During the last 2 minutes, glaze each patty with the remaining 2 tablespoons of teriyaki sauce.

While the patties are cooking, grill the pineapple rings directly over the heat for 1 to 2 minutes on each side until grill marks appear and they caramelize. Alternatively, you can pan-fry them until lightly browned.

In the same skillet or in a separate pan, cook the sliced red onion over medium heat for 5 to 7 minutes, stirring occasionally, until they soften and caramelize. For extra flavor, add a splash of teriyaki sauce during the final minute.

Arrange a cooked patty onto the bottom half of each toasted bun. Add a grilled pineapple ring on top, followed by the caramelized onions, one lettuce leaf, and a slice of tomato. Cap it with the top half of the bun and serve immediately.