In a saucepan, combine soy sauce, mirin, grated ginger, minced garlic, brown sugar, and rice vinegar. Heat over medium-low until it simmers and thickens slightly for about 10 minutes.

In a mixing bowl, mix cold ground beef with 2 tablespoons of the cooled teriyaki sauce and panko breadcrumbs. Form into 4 equal patties.

Preheat grill or skillet to medium-high heat (about 400°F). Cook patties for 4-5 minutes on each side until golden brown and internal temperature reaches 160°F.

In a non-stick skillet, place mounds of grated Parmesan to melt and cook for 2-3 minutes until the edges are golden brown and crispy.

Toast brioche buns until golden brown. Spread teriyaki sauce on the bottom bun, add crispy cheese skirt, the patty, lettuce, tomato, and onion. Cap with the top bun.