In a large bowl, combine the ingredients for the beef burger patty : ground beef, green onions, garlic, fresh ginger, hoisin sauce, soy sauce, egg, panko bread crumbs, sesame seed oil, black pepper. Blend well and form into round patties, about 3 to 4 inches in diameter.
Chill the beef patties for about 30 minutes to firm up and for easier handling.
Preheat the outdoor grill to a medium high. Grill on the middle racks at these cooking times that my son Tim recommended : Rare - 2 to 3 minutes per side; medium 3 to 4 minutes per side; well done 7 to 8 minutes per side. Note: these instructions vary depending on the grill and the person grilling.
Separately, mix the spicy mayonnaise by mixing in a small bowl the following ingredients : mayonnaise and sriracha. Blend well and refrigerate till ready to assemble the burger bun.
Heat or toast the burger buns on the outdoor grill at the last 2 minutes before the burgers are done. Top a slice of cheese on the inside part of the bun (bottom side on grill) for 1 minute, if cheeseburgers are desired.
To assemble : spread the spicy mayonnaise on the inside of the heated burger bun. Stack the filling in this order : burgers, tomatoes, sliced onions, arugula or lettuce greens, top off with bread. Garnish with pickles on the side.
Cook’s comments : My son Tim suggested - place the patty on the preheated grill. When it feels right, flip it ONCE and try not to touch the patties too much or they may crumble. Take off the grill when cooked according to your preference.
Ingredient notes : hoisin sauce is a versatile sweetened soybean paste mixed with flavorings like garlic and vinegar. Use it as a marinade for pork or to cook with stir-fries. (as defined in The Chinese Takeout Cookbook by Diana Kuan).