Preheat oven to 350 degrees F (175 degrees C). Grease a 9x13-inch baking dish with butter.
Cut chickens into wings, drumsticks, thighs, and breast portions. Split breasts down the middle and debone the 4 breast halves. Pound breasts to about 1/4-inch thickness. Place chicken drumsticks, thighs, and wings into prepared baking dish in a single layer, placing chicken pieces as close together as possible. Layer ham, then Swiss cheese over the chicken legs, thighs, and wings. Lay the breasts on top.
Mix together almond meal and olive oil in a bowl until crumbly. Sprinkle mixture over top of breasts.
Bake in oven until a thermometer reads 165 degrees F (75 degrees C) in the center of the the thickest part of a thigh, about 1 hour.