How to peel garlic quickly: To do individual cloves, press the side of a heavy knife on a clove and the peel will loosen. To do a whole head of garlic (or 3 heads), place garlic in a bowl. Nestle a smaller bowl on top and press to crush the head(s) slightly. Then turn over the top bowl and cover the larger one (and garlic) with it. Shake the bowls vigorously and the peels will fall off.
Preheat oven to 325 degrees. Heat a large, heavy-bottomed Dutch oven over medium-high heat. When hot, add the oil. Season 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 second side, then transfer browned chicken to a plate.
Add 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 chicken and some of 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 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 chicken with the gravy spooned over the top and sprinkled with the remaining 1 tablespoon parsley.