Slice the chicken breasts in half (like you would a hamburger bun) to create chicken breasts of equal thickness. Next, slice each breast in half to make cutlets about the size of your palm. You will end up with 8 cutlets in total.
Place the cutlets in a large glass bowl with ¼ cup of salt and one tablespoon of sugar. Fill the bowl with cold water until the chicken are covered entirely with water. Place the bowl in the refrigerator and allow the chicken to brine for 30 minutes, but no longer than one hour.
While the chicken brines, make the white sauce. Place the sauce in the refrigerator to keep cool until needed.
Heat a grill to about 400°F. Oil the grates.
Drain the chicken from the brine and pat dry with paper towels. Place the chicken on the hot grill and cook on each side for about 3 or 4 minutes. Due to their thickness, the cutlets will cook quickly. Keep an eye on them. Toast hamburger buns on the grill at the same time the chicken is cooking.
When done (internal temperature of 165°F), place the chicken on a plate, cover with foil and allow to rest.
Wash and slice lettuce and tomato. Set aside.
To assemble the sandwiches, spread a thick layer of white sauce on the bottom of each hamburger bun, then top with lettuce. Place two cutlets on each sandwich, top with tomato slices, a drizzle of white sauce and the top bun. Enjoy!
This recipe makes four sandwiches.