Place chicken breasts on a large sheet of waxed paper; cover with another sheet of waxed paper. Pound breasts to a 1/4-inch thickness. Remove the top sheet of waxed paper and season with the pepper and salt.
Combine the goat cheese, lemon rind, and apple. Divide apple mixture among the chicken breasts and spread the mixture evenly over half the surface of each breast. Fold each breast in half and press the edges together, forming a pocket.
Sprinkle the flour onto a sheet of waxed paper. In a small bowl, combine the egg and 1 tablespoon water. In a shallow bowl, combine the bread crumbs and Parmesan cheese. Dust each breast lightly with flour and dip into egg mixture, moistening all surfaces. Coat with bread crumb mixture, patting the crumbs gently in place. (The chicken can be prepared to this point several hours ahead; cover and refrigerate. When ready to cook, bring the chicken to room temperature.)
In a large, heavy skillet, heat butter and oil over medium heat. Add chicken; cook until browned and cooked through, about 7 to 8 minutes per side. Drain on paper towels and serve.