Season chicken all over with salt and pepper. In a large pot with a tight-fitting lid over medium-high heat, warm the oil. Working in batches, add the chicken and cook, turning frequently, until well browned, 7-10 minutes. Transfer to a plate.

Pour off all but 2 tablespoons of the fat in the pan. Add the garlic cloves and sauté over medium-high heat until lightly browned, about 3 minutes. Pour in the wine and stir to scrape up the browned bits on the bottom of the pan with a wooden spoon. Sprinkle with the thyme. Return the chicken to the pot. Bring the liquid just to a boil, then immediately reduce the heat to low, cover, and braise until the chicken is tender and opaque throughout, about 45 minutes.

Uncover the pot and check one of the thickest pieces; the meat should be cooked through and falling off the bone. If not, re-cover and continue cooking again for doneness every 10 or 15 minutes.

Transfer the chicken to a platter and cover loosely with foil to keep warm. Set aside a few garlic cloves. Set a fine-mesh sieve over a saucepan and strain the pan juices through the sieve into the pan. Press on the garlic cloves to extract as much liquid and pulp as possible. Bring to a simmer over medium-high heat and season to taste with more salt and pepper.

Transfer the chicken to a platter and pour the cooking liquid over the top. Serve the chicken right away with the bread and reserved garlic cloves for squeezing.