In a large skillet, heat 1 tablespoon of the olive oil. Season the chicken pieces with salt and pepper and cook in a single layer over moderately high heat, skin side down, until the skin is golden and beginning to crisp, about 5 minutes. Transfer the chicken to a plate.

Add the chicken stock, water, and white wine to the skillet and cook over moderate heat, scraping up any browned bits from the bottom. Add the carrot, celery, leek, onion, garlic, bay leaves, clove, and peppercorns and bring to a simmer. Return the chicken pieces to the skillet, skin side up. Cover, reduce the heat to low and simmer until the chicken is cooked through, 20 minutes. Transfer the chicken to a plate and cover loosely with foil.

Strain the cooking liquid into a small saucepan. Bring to a boil, then simmer over moderately high heat until reduced to 2/3 cup, about 20 minutes. Remove from the heat. Whisk in the creme fraiche and butter and season with salt and pepper. Keep warm.

In a medium skillet, heat the remaining 1 tablespoon of oil. Add the chicken, skin side down; cook over moderately high heat until the skin is crisp, 2 minutes. Transfer to plates, ladle the sauce on top and serve.

The chicken can be prepared through Step 2 and refrigerated overnight in the poaching liquid.