Preheat the oven to 325°F. Heat a large, heavy-bottomed Dutch oven over medium-high heat. When hot, add the oil. Season the chicken on both sides with 2 teaspoons salt and 1 teaspoon pepper. Working in batches so as to not crowd the pot, sear the chicken, skin side down, until golden brown, about 6 minutes per batch. Brown briefly on the second side, then transfer the browned chicken to a plate.

Add the garlic to the pot and cook, stirring, until lightly golden, 1 to 2 minutes. Add the lemon juice, Champagne, broth, and thyme, and return the chicken to the pot, nestling the pieces down into the liquid. Make sure some of the garlic is sitting on top of the chicken. Bring the liquid to a boil, cover the pot, and place in the oven. Cook, stirring once midway to ensure even cooking, until the chicken is falling-off-the-bone tender, about 1 hour and 15 minutes.

Transfer the chicken and some of the garlic to a platter, and cover to keep warm. Remove and discard the thyme sprigs.

In a medium bowl, mash the flour and butter together to form a smooth paste. Slowly whisk 1⁄2 cup of the hot juices from the pot into the paste until smooth, then add this mixture to the pot along with 2 tablespoons of the parsley, and whisk to combine. Don’t worry if some of the garlic cloves get smashed—they will help to thicken and enrich the sauce. Cover and cook over medium heat until the gravy has thickened, 10 to 20 minutes longer. Season the sauce with 1⁄2 teaspoon salt and 1⁄4 teaspoon pepper, or more to taste. Serve the chicken with the gravy spooned over the top and sprinkled with the remaining 1 tablespoon parsley.