Step 1Place the chicken breasts and thighs, onion and carrot in a saucepan. Add water to cover and 1 teaspoon of salt. Add the peppercorns. Bring to a boil. Reduce the heat and simmer until the chicken is tender, about 25 to 30 minutes. Remove the chicken from the broth. Strain the broth with a fine mesh strainer. Add water to measure 5 cups. Set aside. Cut the chicken into bite-size chunks. Set aside.
Step 2Melt the butter in a 4-quart saucepan. Cook the diced onion, celery and carrots over medium heat until tender, 5 minutes. Stir in the flour and cook 2 minutes. Add the reserved broth and potatoes. Cover and simmer until the potatoes are tender, about 10 minutes. Stir in the chicken bouillon and rosemary. Season to taste with salt and pepper. Heat to boiling, stirring until thickened, about 2 minutes. Mix in the chicken. Turn into a 13-by-9-inch baking pan. Keep warm while preparing the biscuit dough.
Step 1Heat the oven to 425 degrees.
Step 2Stir together the flour, baking powder, salt, baking soda, sugar, sage and thyme. Cut in the shortening. Stir in the buttermilk just until the dough comes together in a ball.
Step 3Roll out the dough on a floured work surface using a floured rolling pin into a 12-by-8-inch rectangle. With the dull side of a knife, lightly score the top diagonally, forming diamond-shaped biscuits. Do not cut completely through the dough. Carefully lift the dough and place on top of the hot chicken mixture. Brush with the egg white.
Step 4Bake the pie until the biscuit topping is golden brown and cooked through, 20 to 25 minutes.