Make the stock: Pull the turkey carcass apart into smaller pieces; set the meat aside. Put the bones in a large, deep pot and add cold water to cover, 4 to 5 quarts. Add the onion, celery, the 3 quartered carrots and the bay leaf. Tie the parsley and thyme together with twine and add to the pot, then cover and bring to a simmer over medium heat. Uncover, reduce the heat to medium low and cook 3 to 4 hours. Remove the bones and vegetables with a skimmer and discard, then strain the stock though a fine-mesh strainer. Return the stock to the pot and simmer over medium-high heat until reduced by half, 30 to 40 minutes (you'll have about 8 cups stock).
About 45 minutes before serving, prepare the Dumplings. Keep covered with plastic wrap while you make the stew.
Make the stew: Melt the butter in a large, wide pot over medium heat. Add the shallots and cook until soft, about 3 minutes. Add the flour and cook, stirring, 30 seconds. Gradually add the stock, stirring, and bring to a simmer. Season with salt and pepper. Add the sliced carrots and parsnips, cover and cook 5 minutes.
Stir in the turkey meat, lemon juice and green beans. Add the Dumplings in a single layer (leave as squares or pat into rounds). Cover and simmer until the Dumplings are cooked through, about 20 minutes. Ladle into bowls; top with chives.
Whisk the flour, herb mixture, baking powder, baking soda, salt and 1/4 teaspoon pepper. Cut in the butter with a pastry blender until sandy. Stir in the buttermilk.
Turn out onto a floured piece of parchment paper. Pat into a 3/4-inch-thick rectangle.
Cut the dough into rough 2-inch squares with a large knife. Cover with plastic wrap.