In a mixing bowl, combine the ground beef, teriyaki sauce, grated parmesan cheese, salt, and pepper. Mix gently until just combined, being careful not to overwork the meat.
Divide the mixture into 4 equal portions and form each portion into a patty, making a slight indentation in the center to help them cook evenly.
Preheat your grilling pan or outdoor grill over medium-high heat.
Once hot, place the patties on the grill or pan. Cook for about 5 minutes on one side or until browned and release easily, then flip.
Immediately place a slice of American cheese on each patty after flipping. Cook for an additional 3-5 minutes, or until the cheese melts and the patty reaches your desired doneness.
While the burgers are cooking, you may toast the hamburger buns on the grill for about 1 minute, cut side down.
To assemble, spread mayonnaise on the bottom half of each bun, add lettuce leaves, followed by the burger patty, sliced tomato, and the top half of the bun.
Garnish with sliced green onions if desired.