Season the two chicken cutlets with 1/2 teaspoon salt and 1/4 teaspoon pepper. Put 1/4 cup flour in one shallow dish. In a second dish, whisk 2 eggs with 1 tablespoon Parmesan and 1 tablespoon water.
Heat 1 1/2 tablespoons olive oil in a large skillet over medium-high heat. Dredge each cutlet in flour, shake off the excess, then coat in the egg mixture. Cook 3 to 4 minutes per side, until golden and 165°F. Transfer to a plate.
Cook 6 ounces angel hair in well-salted water until just shy of al dente. Reserve 1/4 cup pasta water, then drain.
Reduce the skillet to medium. Melt 1 1/2 tablespoons butter, add 2 minced garlic cloves, and cook 30 seconds. Add 1/4 cup white wine and simmer 1 to 2 minutes. Add 1/2 cup broth, 2 tablespoons lemon juice, 1/2 teaspoon zest, and 1 tablespoon parsley. Simmer 3 to 4 minutes.
Take the skillet off the heat and whisk in the remaining 1/2 tablespoon cold butter. Add 1 to 2 tablespoons reserved pasta water if needed for a glossy, spoonable sauce.
Toss the angel hair with 1 tablespoon butter, 1 small minced garlic clove, 1/2 tablespoon lemon juice, 1 tablespoon parsley, and a splash of pasta water.
Return the chicken to the sauce for about 1 minute. Divide the pasta between two plates, top with chicken, spoon over the sauce, and serve with lemon wedges.