Make the Teriyaki Sauce: In a saucepan, combine soy sauce, mirin, grated ginger, minced garlic, brown sugar, and rice vinegar. Heat over medium heat, stirring occasionally, for about 2-3 minutes until it thickens slightly.

Prepare the Burger Patties: Blend the ground beef with 2 tablespoons of the cooled teriyaki sauce and a handful of panko breadcrumbs. Form into four equal patties and create a dimple in the center.

Cook the Patties: Preheat your grill or skillet to medium-high heat (375°F/190°C). Cook the patties for about 4-5 minutes on one side, flip and cook another 3-4 minutes until they reach an internal temperature of 160°F (71°C).

Create the Cheese Skirt: On a flat surface, sprinkle 1/4 cup mounds of grated Parmesan cheese. Place cooked patties on top of cheese mounds in the last 1-2 minutes of cooking to form the crispy skirt.

Assemble the Burgers: Toast the burger buns lightly. Spread teriyaki sauce on the bottom bun, place the cheesy patty on top, and add lettuce, tomato slices, and red onion.