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Pat chicken pieces dry and season liberally with salt and pepper on both sides
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Heat oil in a 12-inch heavy skillet or casserole (either should have a lid) over medium-high heat.
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Cooking in two batches, place 4 chicken pieces, skin side down, into the oil and sear until golden brown (but do not turn), 3 to 5 minutes, then transfer these to a plate.
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Pour off all but 2 Tbsp fat from skillet. Cook garlic, shallots and celery in skillet over medium heat, stirring frequently, until golden and just beginning to soften, about 5 minutes. Add wine and boil, uncovered, until reduced by about half. .
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Return chicken, this time with the skin side up, to skillet. Place the thyme sprigs around the chicken. Add the water and bring to a quick boil. Then reduce the heat to a simmer.
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Braise, covered partially with the lid, until cooked through, 25 minutes.
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Remove the chicken pieces and celery to a platter and cover loosely with foil to keep warm.
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Bring the remaining liquid back up to a boil and cook, reducing the liquid to about 1/2 to 3/4 cup.
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Plate the chicken and celery with your carb of choice (mashed potatoes, egg noodles, roast potatoes etc) and pour some of the sauce over everything.
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Sprinkle with extra thyme and parsley.