Season 2 whole skinless boneless chicken breasts on both sides with 1 tsp kosher salt.
Melt 1 tbsp unsalted butter and 1 tbsp olive oil in a deep sided skillet and add chicken pieces; cook over medium high heat until golden on both sides, about 5-6 minutes per side. Once golden, remove chicken to a plate and set aside.
Add 1 medium chopped onion , 1 medium carrot, finely chopped and 2 cloves crushed garlic to hot oil and butter and sauté until onions are translucent, about 5 minutes.
Add 2 large tomatoes, 4 sprigs chopped tarragon, 3 sprigs chopped parsley, 3 sprigs chopped thyme, 2 bay leaves and season with 1 tsp black pepper and 1 tsp kosher salt. Then stir to combine.
Add 2 tbsp tomato paste, 1 small can tomato sauce - (8 oz)1/2 cup water, 2 tbsp tarragon vinegar or white wine vinegar, and 1/3 cup dry white wine. Bring to a simmer then cover with the lid propped open with a wooden spoon. Cook until the vegetables are soft and the liquid has reduced by about a quarter, about 20 minutes.
Ladle mixture into a fine sieve set over a mixing bowl and push solids through with the back of a wooden spoon until only the pulp remains.
Return sauce to the pan; the sauce should be velvety smooth.
Return chicken to sauce and heat gently to warm the chicken through, 3-4 minutes over medium heat.