Heat a grill to high. Brush the green onions with some oil and grill for 2-3 minutes per side. Let cool then finely chop. Stir the grilled onions into the mayonnaise along with the garlic clove and salt and pepper to taste.
Combine the beef with 2 tablespoons Mongolian Sauce. Divide the meat into 4 equal portions. Form each portion loosely in a ¾-inch thick patty and, using your thumbs, make a deep depression into the center of each patty. Brush both sides of the burgers with oil and season generously with salt and pepper.
Grill the burgers over high heat for 3-4 minutes or until browned on the bottom and you see juices rising to the top. Flip the burger and grill for another 4 minutes for medium-rare or 5-6 minutes for medium. Remove the burgers from the grill and brush with remaining tablespoon of Mongolian Sauce.
Spread a little of the mayonnaise on the bottom buns, top with lettuce, a burger, and then slather the rest of the mayonnaise on top of the burgers. Top with the roasted red pepper strips, and serve.
If using a grill pan: Heat the grill pan on top of the stove over high heat until a drop of water sizzles immediately when splashed on the pan. Cook as directed above.