To make grilled onions: Place the sliced onions on a microwave-safe plate. Cook for about 5 minutes in the microwave until onions are soft. Coat a large nonstick pan with cooking spray. Add olive oil and heat. Add cooked onions and 1½ tablespoons Worcestershire sauce. Sauté on medium heat until onions are very soft and browned about 6-8 minutes. If the pan gets too dry, add a little water while cooking. Stir often. Keep warm or reheat before serving. These can be made earlier in the day and reheated.
Heat a George Foreman grill or Panini maker.
In a large bowl, add ground beef, egg white, grated onions, Worcestershire sauce, barbecue sauce, seasoning salt, and pepper. Using your hands or a fork, gently mix together. Shape mixture into 12 balls and flatten into very thin patties. Form the patties as large as you can since they shrink when cooked.
To cook on a George Foreman grill or Panini maker: Heat the George Foreman grill or Panini maker and coat with cooking spray. Place burgers on the grill, flatten a bit with a spatula, close lid, and cook for about 2 minutes until cook through. Remove from grill to a plate and continue cooking the remaining patties. Coat with cooking spray each time.
To cook in a pan: Coat a large nonstick pan with cooking spray. Add 4-6 patties and cook over medium-high heat, for 2 minutes. Turn over and continue to cook for 2-3 minutes until cook through. Remove patties from pan. Coat pan with cooking spray and continue cooking the remaining patties. Be sure to coat the pan with cooking spray before cooking all the patties.
To serve: Spread each sandwich thin with 1 teaspoon ketchup and mustard. Add 1 patty and top each with a tomato slice and some grilled onions. Add the top of the sandwich thin.