How to Make It
In a very large enameled cast-iron casserole, melt the butter over medium heat. Season the chicken with salt and pepper. Add the chicken skin side down and cook, turning once, until lightly browned on both sides, 10 to 12 minutes.
Scatter the onion, sliced and whole mushrooms, and garlic around the chicken. Cook, shaking the pan occasionally, until the onion and mushrooms soften slightly, about 4 minutes. Sprinkle the flour on top, then turn the chicken and vegetables to coat them in the flour. Gradually add the beer and bring to a boil. Add the crème fraiche, cover and braise over low heat until the chicken is very tender, about 45 minutes; discard the whole mushroom and garlic skins.
Transfer the chicken to shallow bowls. Season the sauce with salt and pepper. Spoon some of the sauce, onion, and mushrooms over the chicken. Serve.