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1
Put oven rack in middle position and preheat oven to 350F.
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2
Remove and discard any feathers from quail, then remove necks if necessary with poultry shears and discard.
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3
Rinse quail inside and out and pat dry, then sprinkle all over with salt and pepper.
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4
Heat oil and butter in a deep 12-inch heavy ovenproof skillet over moderate heat until foam subsides, then brown quail on all sides, 12 to 15 minutes total.
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5
Transfer to a large plate.
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6
Pour off all but 2 tablespoons fat from skillet, then saute carrot, onion, and thyme over moderately high heat, stirring and scraping up brown bits, 2 minutes.
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7
Return quail (with any juices on plate) to skillet with vegetables and roast, breast sides up, in oven until just cooked through (cut into an inner thigh; meat will still be slightly pink), about 10 minutes.
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8
Transfer quail to a platter and loosely cover with foil.
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9
Add Cognac to vegetables in skillet and boil 1 minute, then add broth and boil until liquid is reduced by half, about 6 minutes.
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10
Stir in creme fraiche and simmer until slightly thickened, 3 to 4 minutes.
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11
Pour sauce through a fine-mesh sieve into a bowl, pressing hard on and discarding solids, then pour back into skillet.
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12
Add grapes and chestnuts and simmer, stirring occasionally, until heated through and grapes turn a shade lighter, about 3 minutes.
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13
Transfer grapes and chestnuts to platter with quail, then boil sauce until thick enough to lightly coat back of spoon, about 10 minutes.
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14
Add salt and pepper to taste, then spoon some sauce over quail and serve remainder on the side.
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15
*Available at specialty foods shops, D'Artagnan (800-327-8246), and Cavendish Game Birds (802-885-1183).
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16
**Available at specialty foods shops and some supermarkets.
