01 - Pat the chicken pieces thoroughly dry with paper towels and season generously on all sides with salt and freshly ground black pepper.
02 - Heat olive oil in a large, heavy-bottomed skillet or Dutch oven over medium-high heat. Sear the chicken pieces skin-side down first, then brown on all sides until deeply golden, approximately 5 to 7 minutes. Transfer the browned chicken to a plate and set aside.
03 - In the same skillet with the rendered drippings, add the finely chopped onion and minced garlic. Cook over medium heat, stirring occasionally, until the onion is translucent and softened, about 3 minutes.
04 - Add the sliced button mushrooms to the skillet. Continue cooking, stirring occasionally, until the mushrooms release their moisture and begin to develop a golden-brown color, approximately 5 minutes.
05 - Stir in the tomato paste and cook for 1 minute to caramelize slightly. Pour in the dry white wine, using a wooden spoon to scrape up any browned bits from the bottom of the pan. Let the wine reduce by about half, roughly 2 minutes.
06 - Add the diced tomatoes and chicken stock to the skillet, stirring to combine. Return the browned chicken pieces to the pan, nestling them into the sauce. Bring the mixture to a gentle simmer.
07 - Add the chopped tarragon, stir gently, then cover the skillet with a tight-fitting lid. Reduce the heat to low and braise for 35 to 40 minutes, until the chicken is fall-off-the-bone tender and cooked through to an internal temperature of 165°F.
08 - If desired, stir in the unsalted butter for extra richness and a velvety finish. Taste the sauce and adjust the seasoning with additional salt and pepper as needed.
09 - Sprinkle generously with freshly chopped parsley just before serving. Present family-style in the skillet or transfer to a warm platter.