Separate the cloves of garlic and drop them into a pot of boiling water for 60 seconds. Drain the garlic and peel. Set aside.
Dry the chicken with paper towels. Season liberally with salt and pepper on both sides. Heat the butter and oil in a large pot or Dutch oven over medium-high heat. In batches, saute the chicken in the fat, skin side down first, until nicely browned, about 3 to 5 minutes on each side. Turn with tongs or a spatula; you don’t want to pierce the skin with a fork. If the fat is burning, turn the heat down to medium. When a batch is done, transfer it to a plate and continue to saute all the chicken in batches.
Remove the last chicken to the plate and add all of the garlic to the pot. Lower the heat and saute for 5 to 10 minutes, turning often, until evenly browned. Add 2 tablespoons of the Cognac and the wine, return to a boil, and scrape the brown bits from the bottom of the pan. Return the chicken to the pot with the juices and sprinkle with the thyme leaves. Cover and simmer over the lowest heat for about 30 minutes, until all the chicken is done. Remove the chicken to a platter and cover with aluminum foil to keep warm.
In a small bowl, whisk together 1/2 cup of the sauce and the flour and then whisk it back into the sauce in the pot. Raise the heat, add the remaining tablespoon of Cognac and the cream, and boil for 3 minutes. Add salt and pepper, to taste; it should be very flavorful because chicken tends to be bland. Pour the sauce and the garlic over the chicken and serve hot.
You can save a ton of time and effort by using 40 cloves of garlic from a jar of pre-peeled garlic cloves.