Season chicken generously with salt. Heat oil in a Dutch oven over medium-high heat. Sear chicken until well browned on both sides, 10 to 12 minutes total. Transfer to a plate.
Reduce heat to medium. Add shallots and garlic and cook until softened, 2 to 3 minutes. Stir in tomato paste and cook for 1 minute.
Add vinegar and wine, scraping up browned bits. Simmer until slightly reduced, 3 to 4 minutes.
Add broth, thyme, and tarragon. Bring to a simmer. Return chicken to pot, skin side up. Cover and cook gently until chicken reaches 165°F/75°C, about 20 minutes.
Meanwhile, boil potatoes in salted water until very tender, 15 to 18 minutes. Drain and mash or rice. Stir in butter and warm milk until smooth. Season with salt.
Transfer chicken to a plate. Stir cream into sauce and simmer until lightly thickened. Return chicken to warm through.
Serve chicken over mashed potatoes, spooning sauce over top.