Place your palm flat against the top of each chicken breast and carefully slice each piece in half horizontally; you'll be left with 4 thinner chicken breast pieces. Place each chicken cutlet between 2 sheets of waxed paper and use the smooth side of a mallet (or a rolling pin) to pound them until they're very thin. Season the chicken on both sides lightly with salt and pepper and set aside.
In a bowl, combine the almond, tapioca flour, 1 teaspoon salt. Stir to combine and set aside.
In a separate bowl, add the egg and water. Whisk until frothy.
Using one hand, dip the chicken breasts in the egg and flip to coat both sides, shake off excess. Using your other hand, then immediately dredge in the almond flour mixture to coat evenly all over. Shake off excess coating. Set on a clean plate and continue until all of the chicken is coated. (Using separate hands for each station prevents your hands getting too caked in the mixture and makes it easier to work with the chicken).
Heat the oil in a large skillet over medium-high heat. When the oil is hot (but not smoking) place the chicken into the skillet and fry until golden brown on each side and cooked through, 4-5 minutes per side.
Transfer the cooked chicken to a plate and immediately serve with the salad.