01 - Pat the chicken thighs and drumsticks thoroughly dry with paper towels. Season both sides generously with salt and freshly ground black pepper.
02 - Heat olive oil in a large, heavy-bottomed skillet or Dutch oven over medium-high heat. Working in batches if needed, brown the chicken pieces on all sides until deeply golden, about 5 to 7 minutes. Transfer the browned chicken to a plate and set aside.
03 - Reduce the heat to medium. Add the finely chopped onion and minced garlic to the same skillet. Cook, stirring occasionally, until the onion is softened and translucent, 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, about 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 half, about 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 - Stir in the chopped fresh tarragon. Cover the skillet with a tight-fitting lid, reduce the heat to low, and braise for 35 to 40 minutes, or until the chicken is fork-tender and cooked through to an internal temperature of 165°F.
08 - If desired, stir in the unsalted butter to enrich the sauce. Taste and adjust the seasoning with additional salt and pepper as needed.
09 - Sprinkle generously with fresh chopped parsley just before serving. Present alongside buttered noodles, steamed rice, or crusty French bread.