Cut each chicken breast in half. Pound all four pieces with a mallet until they are 1/4-inch thick or less. Sprinkle the chicken pieces with salt and pepper.
Remove the butter log from the freezer and cut it in half. Place a half log of butter on one piece of chicken breast, and the other half log on another breast. Fold the edges of the chicken breast pieces over the logs of butter to encase them, and top each breast with the 2 remaining pieces of chicken, to completely enclose the butter.
Place the breadcrumbs in a bowl or shallow baking dish. In another small bowl, beat the eggs with a little water. In a separate small bowl, mix together 1/4 teaspoon pepper and the remaining 3/4 cup flour. Coat the chicken pieces with the seasoned flour, dip them in the egg mixture and roll them in the breadcrumbs. Place the breaded chicken on a shallow tray and chill in the refrigerator for 30 minutes.
Heat a 1/2 inch or so of vegetable oil to medium-high in a deep frying pan. In batches if needed, fry the chicken pieces for about 5 minutes, then flip and fry the other side until the chicken is golden brown, or another 5 minutes.