Add the carrot, parsnip, squash and turnip to the stock and season with salt and pepper. Cover partially and cook over moderate heat until the vegetables are just tender, about 10 minutes. Add the leek and cook until tender. Holding the pastry bag in 1 hand and a table knife in the other, cut 1-inch lengths of the dumpling dough into the simmering soup. When all of the dough has been added, stir gently and cook until tender, about 2 minutes. Add the shredded chicken, cook until heated through and serve.