Cut the chicken breasts into 12 pieces of roughly equal size. Place the pieces between sheets of waxed paper, and pound with a mallet until they're thin. Season with salt and pepper. Place cheese and 4 tablespoons parsley in a wide, shallow bowl. Slowly add the beaten egg, whisking until it's smoothly incorporated. Place the flour on a wide plate. Dip the pounded chicken in the egg mixture. Remove, letting excess egg drip off. Place each cutlet in the flour, and coat lightly. Remove from flour and hold them in a single layer. Add the olive oil to 2 saute pan large enough to hold 3 cutlets in a single layer. Place over medium - high heat. When the oil is hot, add the cutlets. Saute, turning once, until the cutlets are golden on the outside, just cooked on the inside (about 2 minutes per side). Remove the cutlets, and hold them in a single layer. Repeat with remaining cutlets. Spill the oil out of the saute pan.
Return the pan to high heat. Add 1/4 cup of white wine to each pan and reduce it to 2 tablespoons in each pan. Divide the stock and the lemon slices between the pans. Boil for 5 minutes, then remove the lemon slices. Keep boiling the sauce until it's reduced to 1/2 cup. Turn heat to very low. Swirl 2 tablespoons butter into each pan until the sauce is thickened. Add cutlets to each pan, turning them until they are coated in sauce. Divide cutlets among 4 plates, garnish with remaining 4 tablespoons of parsley, and serve immediately.