Preheat the oven to 400 degrees. Line a sheet pan with parchment paper. Place each chicken breast between 2 sheets of parchment paper or plastic wrap and pound out to 1/4-inch thick. Sprinkle both sides with salt and pepper. Mix the flour, 1/2 teaspoon salt, and 1/4 teaspoon of pepper in a shallow plate. In a second plate, beat the egg and 1/2 tablespoon of water together. Place the bread crumbs on a third plate. Dip each chicken breast first in the flour, shake off the excess, and then dip in the egg and bread crumb mixtures. Heat 1 tablespoon of olive oil in a large saute pan over medium to medium-low heat. Add the chicken breasts and cook for 2 minutes on each side, until browned. Place them on the sheet pan and allow them to bake for 5 to 10 minutes while you make the sauce.
For the sauce, wipe out the saute pan with a dry paper towel. Over medium heat, melt 1 tablespoon of the butter and then add the lemon juice, wine, the reserved lemon halves, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Boil over high heat until reduced in half, about 2 minutes. Off the heat, add the remaining 2 tablespoons of butter and swirl to combine. Discard the lemon halves and serve 1 chicken breast on each plate. Spoon on the sauce and serve with a slice of lemon and a sprinkling of fresh parsley.