Place chicken breasts between 2 sheets of plastic wrap or parchment paper; flatten to an even thickness (about 1/2 inch). Sprinkle top of chicken evenly with 1/2 teaspoon salt and 1/4 teaspoon pepper. Sprinkle top of chicken with 1 tablespoon cornstarch; rub gently to coat. Turn chicken over; sprinkle with 1 tablespoon cornstarch, and rub gently to coat.
Heat oil in a large nonstick skillet over medium-high heat. Add chicken to pan; cook until well browned and cooked through, about 5 to 6 minutes on each side. Remove chicken from pan.
Add shallots to drippings in pan; cook 1 minute, stirring constantly. Add wine, scraping pan to loosen browned bits. Cook until wine almost evaporates, about 2 minutes. Add asparagus to pan. Combine stock and remaining 1/2 tablespoon cornstarch, stirring well. Add stock mixture, remaining 1/4 teaspoon salt, and remaining 1/4 teaspoon pepper to pan; cook until sauce slightly thickens and asparagus is crisp-tender, about 3 minutes. Remove from heat. Add butter, capers, and lemon juice, stirring until butter melts. Return chicken to pan; sprinkle with parsley, if desired.