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1
Cook onion, celery, and garlic in butter in a 10-inch heavy skillet over moderate heat, stirring, until pale golden, about 10 minutes. Add demi-glace and boil, stirring occasionally, until most of liquid is evaporated, about 3 minutes. Stir in tarragon, vinegar, walnuts, dried figs, bread crumbs, salt, and pepper, then spread stuffing on a plate to cool.
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2
Heat 2 tablespoons butter in a 12-inch heavy skillet over moderately high heat until foam subsides, then brown figs, cut sides down, without stirring, about 3 minutes. Transfer figs to a bowl with a slotted spatula. Add shallot and celery to skillet and saute, stirring, until golden, 3 to 5 minutes. Add wine and 8 to 10 browned fig halves (reserve remainder) and boil, stirring and mashing figs, until wine is reduced to a syrup, about 5 minutes. Stir in demi-glace and bring to a boil. Stir arrowroot into vinegar until dissolved, then add to skillet, whisking to incorporate. Boil sauce 2 minutes, then pour through a fine-mesh sieve into a 2-quart heavy saucepan, pressing on and then discarding solids. Stir in tarragon, salt, and pepper.
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3
Discard any disposable metal skewers from cavity of each quail, then rinse quail inside and out and pat dry. Stuff 1 quail with a scant 1/4 cup stuffing, pressing and shaping it to fill out breast. Tie legs together with string and push legs up against body. Thread cavity closed with a wooden pick. Repeat with remaining quail.
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4
Put oven rack in middle position and preheat oven to 350u00b0F.
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5
Sprinkle quail all over with salt and pepper. Heat 1 tablespoon each butter and oil in cleaned 10-inch heavy skillet over moderately high heat until foam subsides, then brown 6 quail on all sides, about 10 minutes total. Transfer with tongs to a large shallow baking pan. Wipe skillet clean and brown remaining 6 quail in same manner in remaining tablespoon each of butter and oil.
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6
Remove strings and picks from all quail, then roast quail, breast sides up, until just cooked through (check inner thigh - meat will still be slightly pink), 10 to 15 minutes. Add reserved browned figs to pan for last 2 to 3 minutes of roasting.
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7
While quail roast, return sauce to a simmer, then add remaining 2 tablespoons butter, whisking until incorporated.
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8
Transfer quail and figs to a serving dish and pour any juices from baking pan into sauce. Serve quail with sauce.
