Season the bone-in chicken pieces generously with kosher salt and black pepper, coating all sides for maximum flavor. Prepare russet potatoes for boiling if making garlic mashed potatoes.

In a large skillet, heat butter and olive oil over medium-high heat. Add chicken pieces skin-side down and brown for about 8 minutes. Flip and brown the other side for another 8 minutes. Remove the chicken and keep warm.

In the same skillet, reduce heat and add olive oil if needed. Sauté cremini mushrooms for 3 minutes, then add shallots and garlic, cooking for 1 additional minute.

Deglaze the skillet with chicken stock, stirring in tomato paste and diced Roma tomatoes, followed by beef and chicken stock. Let simmer for 5–7 minutes.

Stir in sour cream and fresh tarragon. Return the chicken to the skillet, covering it with sauce, and simmer for 5 minutes.

Plate the Classic French Hunter's Chicken, garnished with parsley, and serve hot over garlic mashed potatoes or rice.