Pound the chicken breasts until they are 1/4-inch thick. You can use either a meat mallet or a rolling pin.
Combine the flour, salt, and pepper on a dinner plate. On a second plate, beat the eggs with 1 tablespoon of water. On a third plate, combine the bread crumbs and 1/2 cup grated Parmesan. Coat the chicken breasts on both sides with the flour mixture, then dip both sides into the egg mixture and dredge both sides in the bread-crumb mixture, pressing lightly.
Heat 1 tablespoon of butter and 1 tablespoon of olive oil in a large saute pan and cook 2 or 3 chicken breasts on medium-low heat for 2 to 3 minutes on each side, until cooked through. Add more butter and oil and cook the rest of the chicken breasts. Toss the salad greens with lemon vinaigrette. Place a mound of salad on each hot chicken breast. Serve with extra grated Parmesan.
Lemon Vinaigrette:
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