Pat the chicken dry with paper towels and sprinkle both sides liberally with salt and pepper. Place 1/2 cup flour in a bowl and dredge the chicken in the flour. In a large (12-inch) ovenproof pot such as Le Creuset, heat the oil. Add the chicken in three batches (don't crowd them!) and brown lightly over medium-high heat for 3 to 5 minutes on each side. Remove the chicken to a plate and continue until all the chicken is browned.
Add the whole garlic cloves, mushrooms, and thyme to the pot and cook over medium heat for 5 minutes, stirring occasionally. Add the sherry and cook for 1 minute, scraping up the brown bits. Add the minced garlic and cook for 2 more minutes. Add the wine, chicken stock, 1 tablespoon salt, and 1 teaspoon pepper and bring to a simmer. Add the chicken (large pieces first), cover, and place in the middle of the oven for 30 to 35 minutes, until the chicken is cooked through (about 165 degrees on an instant-read thermometer).
Remove the chicken to a bowl and discard the thyme. With a fork, mash together the butter and 1/4 cup flour and add it to the sauce. Simmer, stirring constantly, over medium heat for 5 minutes, until slightly thickened. Season to taste (it should be highly seasoned), put the chicken back in the sauce, and serve hot.
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