How to Make It
In a large bowl, combine the wine with the leek, onion, carrots, garlic, parsley, thyme and bay leaves. Put the short ribs in 2 large resealable plastic bags and pour half of the marinade into each bag. Seal the bags, transfer them to a baking sheet and refrigerate overnight.
Preheat the oven to 300°. Remove the short ribs from the marinade and pat dry with paper towels. Using a slotted spoon, remove the vegetables from the marinade.
In a large skillet, heat the vegetable oil. Season the short ribs with salt and pepper and dust with flour, shaking off the excess. Add half of the ribs to the skillet and cook over moderately high heat until browned and crusty, about 3 minutes per side. Transfer the ribs to a large roasting pan. Repeat with the remaining ribs, lowering the heat if the ribs get too dark.
Add the vegetables to the skillet and cook over moderately high heat until they start to brown, about 4 minutes; transfer to the roasting pan. Add the marinade to the skillet and bring to a boil, skimming the surface. Add the chicken and beef stock and bring to a boil, scraping up any browned bits from the bottom of the skillet. Pour the liquid over the ribs. Cover the pan with foil and braise the ribs in the oven for about 3 hours, until the meat is very tender.
Remove the ribs from the roasting pan. Strain the braising liquid and skim off the fat. Discard the vegetables.
In a medium saucepan, heat the oil. Add the shiitake mushrooms and season with salt and pepper. Cover and cook over moderate heat, stirring a few times, until browned, about 5 minutes. Add the shallots and thyme and cook until the shallots are softened, about 4 minutes. Add the garlic and cook until fragrant, about 1 minute. Add the wine and boil over high heat until reduced to 3 tablespoons, about 2 minutes. Add the skimmed braising liquid and boil over moderately high heat for 5 minutes; season the sauce with salt and pepper. Stir in the 2 tablespoons of parsley.
Preheat the oven to 350°. In a large saucepan of boiling salted water, cook the vegetables separately until tender: 3 minutes for the onions; 4 minutes for the turnips; 5 minutes for the carrots and 12 minutes for the potatoes. Trim and peel the onions. Slice the potatoes lengthwise.
Remove the bones from the rib meat and return the meat to the roasting pan. Add the cooked vegetables and pour the sauce on top. Cover with foil and bake for about 15 minutes, until heated through. Serve the stew in shallow bowls, garnished with parsley.