Combine the chicken broth, carrot, celery, onion, bay leaf, and garlic in a small saucepan; bring to a boil. Cook 10 minutes or until reduced to 1¼ cups. Remove solids with a slotted spoon; discard.
Cut the 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, then dredge the chicken evenly in flour.
Heat a large skillet over medium-high heat. Add 1 tablespoon of oil and 1 tablespoon of butter to the skillet. Swirl until the butter melts.
Add 4 cutlets to the pan, then cook 2-3 minutes on each side or until golden. Remove from the skillet to a plate; keep warm loosely covered with a foil tent.
Repeat the procedure with the remaining 1 tablespoon of oil, 1 tablespoon of butter, and 4 cutlets. Set the cutlets aside on a plate; keep warm loosely covered with a foil tent.
Return the large skillet to medium-high heat. Add mushrooms and onions; cook 5 minutes or until browned, stirring occasionally.
Add the chicken broth mixture; bring to a boil, scraping the skillet to loosen browned bits.
Combine 1 tablespoon of cold water and cornstarch in a small bowl.
Stir the cornstarch slurry into the broth mixture, and cook 1 minute or until the 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 its juices back into the pan and smother in the sauce.
Serve each piece of chicken with the sauce drizzled over the top. Enjoy.