Cut the chicken into eighths with a knife or poultry shears, leaving most of the bones and removing only the back and ribs. (You can also have your butcher or meat counter cut the chicken for you or just buy parts.)
Peel the onion, leaving the root intact. Cut in half. Prepare the bouquet garni.
Season the chicken pieces evenly with salt and pepper.
Preheat a heavy-bottomed, 12-inch (30 cm) skillet with a lid over medium. Add the oil and the 2 tablespoons (28 g) butter. Sauté the chicken pieces, skin-side down, for 6 to 8 minutes. The skin should only be slightly browned.
Turn the chicken pieces over. Add the white wine, chicken stock, bouquet garni and onion. Simmer gently, covered, for 30 to 35 minutes.
Knead together the softened butter and the flour (beurre manie).
When the chicken is fully cooked (internal temperature should be at least 165°F [74°C]), remove the pieces from the skillet to a serving platter. Discard the bouquet garni and the onion.
Bring the sauce to a boil and reduce to 1 cup (235 ml). Whisk in the beurre manie. Add the cream and bring back to a simmer for 3 to 5 minutes. Whisk to avoid scorching. Stir in the tarragon, pour over the chicken, and serve.