Step 1Heat oil in large oven-proof skillet over medium-high heat until oil shimmers. Salt and pepper duck legs and wings on both sides, then, working in batches, brown duck, about 5 minutes per side. Set aside. Repeat until all duck has been browned.
Step 2Pour off all but enough fat to coat skillet. Add onion, carrots, celery and a little salt. Cook over medium heat, stirring occasionally, until vegetables soften and begin to brown, about 15 minutes. Add garlic and sprigs of thyme and tarragon. Cook until vegetables are almost tender, 3 to 5 minutes more.
Step 3Arrange duck, skin-side up, over vegetables. Add enough stock to cover vegetables but not duck. Bring stock to simmer.
Step 4Add apples and transfer skillet to 350-degree oven. Gently simmer--just an occasional bubble--basting occasionally, until duck is very tender and well browned, about 2 hours.
Step 5Remove duck from pan and cover loosely with foil. Strain braising liquid, pressing vegetables and apples with wooden spoon. Discard vegetables and apples and return liquid to skillet. Bring to simmer and skim off fat.
Step 6Whisk butter and chopped thyme and tarragon into braising liquid. Return duck to skillet, heat through, then serve.