Bring a medium saucepan of salted water to a boil. Add the pearl onions and cook for 10 minutes. Drain and set aside.
Over a high flame, heat 3 tablespoons of the vegetable oil in a large Dutch oven. Add the mushrooms and saute until they’re soft and brown, about 5 minutes. Reduce the heat and add the leek and ¼ teaspoon of salt. Cook for 5 minutes, stirring frequently, until soft but not brown.
Add the wine and bring to a boil. Reduce the liquid by slightly more than half, about 12 to 15 minutes. Add the chicken jus and the blanched pearl onions and simmer for 20 minutes. Stir in the cream and keep at a low simmer.
Preheat the oven to a whopping 450°F Season the chicken pieces with the remaining 1 teaspoon of salt and ½ teaspoon of black pepper. Heat the remaining 2 tablespoons of vegetable oil in a large ovenproof saute pan over a medium-high flame. When the oil begins to smoke, add the legs and thighs to the pan, skin side down. Brown for 10 minutes, forming a golden crust. Transfer the pan to the oven. Turn the pieces after 5 minutes, and cook for 5 minutes more. Place the chicken pieces, skin side up, in the simmering sauce.
Now add the breasts, skin side down, to the large saute pan, which will be quite hot. Brown the breasts for 5 minutes on each side, then tranfer to the oven for 10 minutes (no need to turn them over). Remove from the oven and add the breasts to the sauce. Turn off the flame and add a few grindings of pepper, the parsley, and the thyme. Serve in large bowls with Spaetzle, egg noodles, or rice.