Step 1

In a 12-inch skillet, melt 2 Tbs. of the butter over medium heat. Add the onions, sprinkle with a couple of big pinches of salt and a few grinds of pepper, and stir to coat the onions. Cover, reduce the heat to medium low, and continue to cook, stirring occasionally, until the onions are tender and lightly browned, about 20 min. Scrape them into a small bowl and set the skillet over medium-high heat. Add 1 Tbs. of the vinegar and stir with a wooden spoon to dissolve any browned bits on the bottom of the pan. Pour the vinegar into the onions and set the skillet aside.

Step 2

If using chicken parts, cut each breast crosswise into two equal-size portions and trim any excess fat or skin from the thighs. Rinse and pat dry.

Step 3

Spread the flour in a pie plate, and season the chicken pieces with salt and pepper. Set the skillet over medium-high heat and add the olive oil and the remaining 1 Tbs. butter. While the butter melts, dredge half of the seasoned chicken pieces in the flour, shaking off the excess. Set them skin side down in the skillet. Brown, turning once, until the skin is crisp and the chicken is evenly browned, 6 to 8 min. total. Lower the heat if the chicken or the drippings threaten to burn. Transfer the chicken pieces to a pan or platter and repeat with the remaining chicken.

Step 4

When all the chicken is browned, pour off all of the fat. Return the skillet to medium-high heat, add the wine, and scrape the bottom of the pan with a wooden spoon to dissolve the drippings. Add the remaining 2 Tbs. of vinegar, the sautéed onions, and 1 tsp. of the tarragon. Return the chicken pieces, skin side up, to the skillet, arranging them in a single snug layer. Partially cover, leaving a small gap for the steam to escape, and lower the heat to maintain a low simmer. Continue to simmer gently, turning every 10 min., until the chicken is tender and cooked through, about 30 min. total.

Step 5

Transfer the chicken to a platter. Increase the heat to a more rapid simmer, and stir in the crème fraîche (or cream); the sauce may appear broken at first, but it will come together. Taste for salt and pepper. Add the remaining 1 tsp. tarragon and spoon over the chicken to serve.