Make the garlic butter: Mix the softened butter, minced garlic, chopped parsley, salt, and pepper in a small bowl. Form the mixture into a rectangle on plastic wrap, roll it up, and put it in the freezer for 15 minutes.
Prepare the chicken: Put each chicken breast between two pieces of plastic wrap. Use a meat mallet to gently pound the chicken until it's about ¼ inch thick.
Add the butter: Take the butter from the freezer and cut it into 4 pieces. Put one piece of butter in the center of each chicken breast. Fold the sides of the chicken over the butter, then roll it up like a burrito. Make sure the butter is completely covered.
Coat the chicken: Set up three shallow bowls - one with flour, one with beaten eggs, and one with breadcrumbs. Roll each chicken in flour, then dip in egg, and finally coat with breadcrumbs. Press the breadcrumbs gently to help them stick.
Chill: Put the coated chicken in the refrigerator for at least 30 minutes to help everything stay together.
Preheat: Turn on your air fryer and set it to 375°F. Let it heat up for 3 minutes.
Cook: Spray the air fryer basket with cooking spray. Place the chicken pieces in the basket, making sure they don't touch. Spray the tops of the chicken with cooking spray. Cook for 10 minutes, then flip the chicken, spray again, and cook for another 5-7 minutes until golden brown and the internal temperature reaches 165°F.
Rest: Let the chicken rest for 5 minutes before serving.