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. 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. 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. Put oven rack in middle position and preheat oven to 350°F. 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. 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. While quail roast, return sauce to a simmer, then add remaining 2 tablespoons butter, whisking until incorporated. Transfer quail and figs to a serving dish and pour any juices from baking pan into sauce. Serve quail with sauce.