Wash the chicken and pat dry with a paper towel. In a shallow dish, combine the chicken and the teriyaki sauce. Cover the dish and let marinate in the refrigerator for 2 hours, or up to overnight. Add the olive oil to a large skillet and heat over medium-high heat. Cook the chicken on each side for about 3 minutes. Split the chicken breast with a fillet knife about 3/4 through the center. Place the inside of each breast down into the pan. Place a slice of cheese on top of each chicken breast and cook for 3 minutes or until chicken is fully cooked through.
Toast the buns in the skillet or a toaster. Coat each bun with the Secret Sauce. Stack the lettuce, sliced tomato, and onion on the bottom buns. Add the chicken and top buns and serve immediately.
Combine the ketchup and mayonnaise, to taste, in a small bowl. The sauce should be not too pink and not too red.