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.