Place first 6 ingredients in a saucepan; bring to a boil. Cook 6 minutes or until reduced to 1 1/4 cups. Remove solids with a slotted spoon; discard.
Cut chicken breast halves in half horizontally to form 8 cutlets. Sprinkle evenly with 1/4 teaspoon salt and 1/4 teaspoon pepper. Place flour in a shallow dish; dredge chicken in flour. Heat a large skillet over medium-high heat. Add 1 tablespoon oil and 1 teaspoon butter to pan; swirl until butter melts. Add 4 cutlets to pan; cook 2 minutes on each side or until golden. Remove from pan; keep warm. Repeat procedure with remaining 1 tablespoon oil, 1 teaspoon butter, and 4 cutlets.
Return skillet to medium-high heat. Add mushrooms and onions; cook 5 minutes or until browned, stirring occasionally. Add stock mixture; bring to a boil, scraping pan to loosen browned bits. Combine 1 tablespoon water and cornstarch in a small bowl. Stir cornstarch mixture into stock mixture; cook 1 minute or until sauce thickens. Stir in remaining 1/4 teaspoon salt, remaining 1/4 teaspoon pepper, and tarragon. Spoon sauce over chicken.