For the chicken and filling:
Wipe the chicken pieces dry. Put the butter in a wide 5 quart Dutch oven and turn on the heat to medium. When it is melted and hot, add the chicken and brown it well on all sides, turning occasionally, about 5-10 minutes. Remove but reserve the fat.
Cut 1 onion into quarters and add it to the chicken with the thick-sliced carrot and celery. Add the broth and bring it to a simmer. Reduce the heat to a slow simmer, loosely cover, and cook until the chicken is tender, about 30-45 minutes. Remove the chicken, let it cool, then skin, bone, and cut it into bite-sized pieces. Strain and reserve the broth, discarding the solids, and wipe out the pot.
Position a rack in the center of the oven and preheat the oven to 375 degrees F. Put 3 tablespoons of the reserved fat and the ham in the pot where the chicken was cooked. Saute, stirring often, until it loses its raw red color, then add the diced onion, celery, and carrot. Saute, stirring, until softened and beginning to color, about 5 minutes. Add the leek, sage, and thyme and saute until the leek is wilted but not colored, about 2 minutes.
Sprinkle in the flour and cook, stirring constantly, until bubbly and smooth. Slowly stir in 3 cups broth and simmer until thick, about 4 minutes. Stir in the cream and bring it back to a simmer, seasoning with salt (if needed), pepper and a fresh grating of nutmeg.
Bring to a simmer and simmer until thickened, then add the diced chicken. Bring it back to a simmer and turn off the heat. Taste and adjust the seasonings and transfer to a 3-quart casserole, or 4 individual potpie dishes or 10-ounce ramekins.
Put the dough on a lightly floured work surface and roll it out 1/8-inch thick. Cut the dough into a round or rounds a little larger than the casserole or dishes that you are using, then cut out vent holes with a knife or small decorative cookie cutter and put the pastry over the filling. Excess pastry can be cut into decorative shapes for ornament; brush the bottoms brushed with water before placing them. Brush the pastry with the beaten egg white and water. Bake in the center of the oven until the pastry is risen and golden and the filling hot through, about 30 minutes.
For the pastry:
Makes enough to make 2 9-inch pie shells, 1 double crust pie, or to cover 4 individual potpies.
Sift or whisk together the flour, salt, and sage. Cut in the shortening and butter with a pastry blender until the flour resembles coarse meal with random lumps of fat no larger than small peas. Stir in 1/4 cup of ice water and work it in. For a more distinctive chicken flavor, use cold broth rather than ice water. Continue adding water/broth by spoonfuls as needed until the dough is holding together but not wet.
Gather pastry into two balls, press each one into a 1-inch thick flat disk, and cover with plastic wrap. Refrigerate at least 30 minutes or for up to 2 days. Let come to room temperature before rolling out.