Step 1In a large, nonreactive bowl, combine the lamb with the onion, carrot, celery, white wine, garlic, thyme and bay leaf. Cover and marinate, refrigerated, for 24 hours.
Step 2Strain, reserving the wine. Separate the lamb, herbs and wine from the vegetables and reserve each individually. Heat the oven to 300 degrees.
Step 3Place a medium, oven-safe casserole (with a lid, to be used later) over medium-high heat on the stove top. Add the olive oil, then the lamb and sear the pieces on all sides to a rich brown, 8 to 10 minutes. Remove the lamb from the casserole and set aside.
Step 4Add the vegetables and cook until tender, 5 to 7 minutes. Pour in the reserved wine and deglaze the casserole, scraping up the flavoring at the base.
Step 5Return the lamb to the casserole and add the chicken broth, lamb fat, duck fat, the reserved herbs and the sea salt, stirring to combine.
Step 6Bring to a slow boil over high heat, then reduce to a simmer, cover the casserole with the lid and place it in the oven. Braise the lamb until it falls apart at the touch, about 2 1/2 hours.
Step 7Remove the lamb, vegetables and herbs from the casserole, leaving the liquid. Set aside the lamb, discarding the vegetables and herbs. Place the casserole over high heat on the stove top and reduce the liquid by half. Remove from heat and set aside one-half cup of the reduced liquid to make the rillettes, discarding the rest or reserving it for another use.
Step 8Remove the meat from the lamb bones and finely shred. Set aside the shredded meat, discarding the bones. You will have about 1 pound.
Step 1In a medium bowl, combine the lamb, parsley, chives, tarragon, thyme, shallots and duck fat, using your hands or a fork to thoroughly combine.
Step 2Stir in the butter and reduced braising liquid until incorporated. Season with one-half teaspoon salt and several grinds of black pepper, or to taste.
Step 3Serve immediately, or cover and refrigerate until needed; allow the rillettes to come to room temperature before serving.