Place chicken breasts between 2 sheets of heavy-duty plastic wrap; flatten to 1/2-inch thickness, using a meat mallet or rolling pin. Sprinkle chicken evenly with 1/2 teaspoon salt.
Combine 1/4 cup flour, 1/2 teaspoon salt, and 1/4 teaspoon pepper in a shallow dish; dredge chicken in flour mixture, shaking off excess. Dip chicken in milk; dredge in breadcrumbs.
Pour olive oil to a depth of 1/4 inch in a large skillet. Fry chicken, in batches, in hot oil over medium-high heat 5 to 6 minutes on each side. Remove from skillet, and drain on paper towels, reserving 2 tablespoons drippings in pan.
Sauté garlic in hot drippings 20 seconds; add mushrooms, and sauté 3 minutes or until lightly browned. Add green onions, parsley, and capers; sauté 1 minute. Stir in chicken broth, next 3 ingredients, and remaining 1/2 teaspoon salt. Bring to a boil over medium-high heat, and cook, stirring constantly, 2 minutes or until slightly thickened. Serve mushroom sauce over chicken. Serve with spaghetti topped with tomato pasta sauce, if desired.