Place each chicken breast on your work surface and place your palm on top. Slice the chicken in half running the knife flat underneath your hand to form 2 thinner chicken cutlets.
Put the chicken slices in between two pieces of parchment paper. Use a kitchen mallet or the bottom of a mug to pound the chicken thin.
Fill one shallow bowl with the almond flour, another with the eggs, and a third with the breadcrumbs. One by one, dip the chicken first in the almond flour to coat, then in the eggs, and finally in the breadcrumbs, shaking off the excess after each dip. Place them on a baking sheet or piece of parchment while you finish the remaining pieces.
Preheat the olive oil in a large frying pan over medium heat. Add 2 of the pieces of chicken and cook about 3-5 minutes to brown. Flip and brown the other side. Remove to a paper towel lined plate or a cooling rack and repeat with the remaining chicken, adding more oil if needed.
In another small pan, melt the butter over medium low heat. Add in the lemon juice, capers, and parsley and whisk to combine.
Serve the chicken and pour the caper mixture over the chicken pieces.