Combine the flour, 1 teaspoon salt, and 1/2 teaspoon 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, the 1/2 cup grated Parmesan cheese, the lemon zest and thyme, and set aside.
Pound the chicken breasts between two sheets of parchment paper until they are 1/4 inch thick. You can use either a meat mallet or a rolling pin.
Coat the chicken breasts on both sides with the prepared 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 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.
Place the arugula in a large bowl. In a 1-cup glass measure, whisk together the 1/4 cup lemon juice, 1/2 cup olive oil, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Pour enough dressing on the arugula to moisten. Toss well.
Pile some arugula on top of each hot chicken breast. With a very sharp knife or a vegetable peeler, shave the chunk of Parmesan into large shards and arrange them on top of the arugula.