Nonalcoholic sparkling cider gives this dish a sweet-tart flavor.
Cut wing tips from duck below the first joint. Trim flesh from wing bone and discard. Remove excess skin and fat from duck. Save bones for stock.
Prick skin with a fork. In a large, deep, heavy skillet, saute duck over medium heat, skin side down, until most of the fat rendered and skin is golden brown, about 15 minutes. Drain and discard fat from pan periodically. Turn and cook 5 more minutes. Transfer duck to a plate. Drain remaining fat from pan.
Add shallots to skillet and cook over low heat for 2 minutes, scraping brown bits. Add rosemary, 4 sprigs thyme, and turnips. Cook 2 minutes more. Add sugar, cider, stock, and salt and pepper to taste. Bring to a boil. Return duck to pan, reduce heat, cover, and simmer for 10 minutes. Remove breasts and continue cooking legs for 25 more minutes, or until tender.
Remove legs and turnips. Strain; skim and discard fat. Pour sauce into a clean skillet.
In a medium pan, heat oil over high heat. Add mushrooms; cook until soft, about 5 minutes. Season with salt and pepper.
Add mushrooms, turnips, and legs to sauce. Add vinegar and remaining thyme. Bring to a boil and cook over medium heat until sauce is thick enough to coat a spoon, about 5 to 10 minutes, turning legs once or twice. Add breasts just to heat through, and serve.