Combine the chicken broth, carrot, celery, onion, bay leaf, and garlic together in a saucepan; bring to a boil. Cook 10 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 4 cutlets then cut them in half. Season both sides of the chicken pieces with sea salt and freshly cracked pepper, to taste. 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 2-3 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. Set the cutlets aside on a plate loosely covered by a tin foil tent.
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. Taste and season with sea salt and freshly cracked pepper, to taste. Add the fresh tarragon mix until evenly combined. Place the chicken and it's juices back into the pan and smother in the sauce. Enjoy.