In a large mixing bowl, combine the ground beef, 1/4 cup of teriyaki sauce, diced onion, panko breadcrumbs, ginger paste (if using), crushed garlic, salt, and pepper. Mix until just combined, ensuring the ingredients are evenly distributed without overworking the meat. Divide the mixture into four portions and shape each into a patty. Gently flatten the patties to match the size of the hamburger buns, then press a small indentation into the center of each patty to promote even cooking.

Preheat your grill to medium-high heat and lightly oil the grates to prevent sticking. Toast the inside of the sesame seed buns on the grill for about one minute until golden brown. Grill the red onion slices for approximately five minutes until they become tender, then set aside.

Place the patties on the grill and cook for 5 to 7 minutes, flipping halfway through. During the last minute of grilling, brush each patty with the remaining teriyaki sauce to enhance the flavor and add a glossy finish. Check for doneness using an instant-read thermometer, ensuring it reaches at least 160°F (70°C).

To assemble the burgers, place each patty on a toasted bun, drizzle with additional teriyaki sauce, or serve it on the side. Add any preferred toppings such as mayo, lettuce, or cheese, and enjoy.