Adjust the rack to the lower-middle position then preheat the oven to 325 degrees.
Heat the olive oil in an OVEN PROOF large skillet over medium-high heat. Season the chicken thighs with sea salt and freshly cracked pepper, to taste. Place the chicken, skin side down, in the hot skillet and cook for 8 minutes, until well browned. Flip the chicken over and cook for 3 minutes. Remove the chicken from the skillet and set it aside on a plate.
Remove all but 2 tablespoons of fat from the skillet. Add the shallot and garlic and cook, stirring constantly, for 1 minute.
Add the chicken broth, white wine, and red wine vinegar and bring to a simmer, scraping up all the browned bits from the bottom and sides of the skillet.
Nestle the chicken (and any juices) back in the skillet, skin side UP. Place into the oven and bake, uncovered, until chicken registers 195 degrees, about 35-40 minutes.
Using tongs, remove the chicken from the skillet and place it on a serving plate with a loose tin foil tent on top.
Place the skillet over high heat and whisk in the tomato paste and bring to a boil. Cook, stirring and scraping the sides of the skillet until the sauce is thickened and reduced to 1¼ cups, about 5-7 minutes.
Off heat, whisk in the cold butter, chopped tarragon, and up to 1 teaspoon of red wine vinegar. Taste & season with sea salt and freshly cracked pepper, to taste. Pour the sauce around the chicken. Serve and enjoy.