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