Preheat oven to 450 degrees. On a rimmed baking sheet, toss the asparagus with 1 tablespoon olive oil, 1/4 teaspoon salt and 1/4 teaspoon pepper and the lemon zest. Roast the asparagus for 5 to 6 minutes, until just crispy-tender. Transfer to a plate and set aside.
In a small bowl, whisk together buttermilk, mayonnaise, sour cream, garlic, tarragon, lemon juice, 1/4 teaspoon salt and 1/4 teaspoon pepper. Set aside.
Between pieces of plastic wrap or waxed paper, place each chicken breast smooth side down; gently pound with flat side of meat mallet or rolling pin until about 1/4 inch thick. Season chicken with 1/2 teaspoon salt and 1/4 teaspoon pepper.
In shallow pan, stir together flour, remaining 1/4 teaspoon salt and 1/4 teaspoon pepper. Coat both sides of chicken with flour mixture. Beat eggs in shallow bowl with whisk to blend. Place bread crumbs with grated Parmesan in another shallow bowl. Dip chicken into beaten eggs; turn to coat. Dredge in bread crumbs/Parmesan, coating completely.