Using a meat mallet, pound the chicken breasts until they are 1/4 inch thick.
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 cheese. 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 sauté pan. Cook 3 chicken breasts on medium-low heat for 2 to 3 minutes on each side, until cooked through and golden. Remove from pan and set aside, covered to keep them warm. Add more butter and oil to the pan, and cook the rest of the chicken breasts.
Make the vinaigrette in a small bowl by whisking together the juice, olive oil, salt, and pepper. Toss the salad greens with vinaigrette.
To serve, place a mound of salad on each plate and then place a chicken breasts on each salad. Shredded additional Parmesan cheese on top.