Cut each chicken breast half in half crosswise and pound between plastic wrap to 1/4-inch thickness; Melt butter with 2 tablespoons oil in a large nonstick skillet over medium heat.
Combine flour, 1/4 teaspoon salt and pepper in a plastic or paper bag. In batches, add chicken to bag; shake to coat lightly with flour mixture. Dip in eggs to coat both sides, letting excess egg drip off. Discard any extra flour mixture and egg. Sauté chicken in skillet 2 to 3 minutes per side or until golden brown; transfer to a plate and set aside. Add sherry to skillet; cook, stirring to scrape up browned bits, until reduced to about 1 tablespoon. Pour over chicken.
Add remaining oil and garlic to same skillet over low heat; cook 2 minutes or until garlic is just beginning to brown. Add hot cooked pasta, 2 tablespoons parsley, remaining 1/4 teaspoon salt and pepper flakes; toss well. Transfer mixture to a shallow 1-1/2 quart-baking dish. Arrange chicken over pasta; top with cheese. Bake in a preheated 400°F oven 10 minutes or until heated through and cheese is melted. Top with remaining parsley, if desired.