In a saucepan bring broth to a boil. Add carrots, potatoes, and celery, and simmer for 10-15 minutes, or until veggies are tender. Transfer veggies to a bowl, reserving the broth, and add the chicken to the bowl. In another saucepan, cook the onion in the butter over mod-low heat, stirring, until it is softened. Add flour and cook the roux, stirring, for 3 minutes. Add 3 cups of broth in a stream, whisking. Bring mixture to a boil, whisking. Add the thyme and simmer the sauce, stirring occasionally, for 5 minutes. Stir in nutmeg, parsley, and salt and pepper to taste. Pour the sauce over the chicken and veggies. Stir the mixture gently, until it is just combined. Transfer the mixture to a 2-qt. Baking dish. Into a bowl, sift together the flour, baking powder, baking soda, and salt. Add the butter and shortening and blend the mixture until it resembles meal. Add the cheese and toss. Break egg into a measuring cup and add buttermilk to it so that the total volume is 1/2 cup. Beat with a fork. Add to the flour mixture, stirring until the mixture just forms dough. Gather dough into a ball and, on a floured surface, pat it out 1/2-inch thick. Cut as many rounds as possible using a 2-inch fluted biscuit cutter, dipped in flour. Gather the scraps and pat and cut again. Arrange the rounds on the chicken mixture. Beat the egg yolk with the milk and brush the tops of the biscuits with the wash. Prick biscuits with a fork. Bake in the middle of a preheated 450f oven for 15-25 minutes, or until biscuits are puffed and golden, and the filling is bubbling.