How to Make It
Make the crust In a food processor, pulse the flour with the sugar and salt. Add the butter and schmaltz and pulse until the mixture resembles a coarse meal. Drizzle the buttermilk over the top and pulse until the dough just starts to come together. Turn out onto a work surface, gather any crumbs and gently knead the dough 2 to 3 times until smooth. Divide in half and flatten each piece into a disk. Wrap in plastic and refrigerate until firm, 1 hour.
Preheat the oven to 375°. On a floured work surface, roll out 1 disk of dough to a 12-inch round, about 1/4 inch thick. Ease the round into a 9-inch deep-dish pie plate. Trim the overhang to 1/2 inch. Line the crust with parchment paper and fill with pie weights or dried beans. Bake for about 20 minutes, until the crust is just set. Remove the parchment paper and bake for 5 to 7 minutes longer, until lightly browned. Let cool completely.
Meanwhile, make the filling Season the chicken with salt and pepper. In a large saucepan, heat 2 tablespoons of the oil until shimmering. Add the chicken and cook over moderately high heat, turning occasionally, until browned, about 7 minutes. Add the chicken stock and bring to a boil. Simmer over moderately low heat, turning the chicken occasionally, until an instant-read thermometer inserted in the thickest piece registers 160°, about 15 minutes. Using tongs, transfer the chicken to a plate and let cool. Boil the stock over high heat until reduced to 1 cup, 15 to 20 minutes more; transfer the stock to a small heatproof bowl. Discard the chicken skin and bones, then cut the meat into 1-inch pieces. Wipe out the saucepan.
In the saucepan, melt the butter in the remaining 2 tablespoons of oil. Add the onion, carrot, celery, garlic and a generous pinch of salt. Cook over moderate heat, stirring occasionally, until softened, about 10 minutes. Add the chicken and flour and cook, stirring, until the vegetables and chicken are evenly coated. Gradually stir in the milk and reduced stock and bring to a boil. Simmer over moderately low heat, stirring occasionally, until thickened, 5 to 7 minutes. Stir in the parsley, chives, thyme and vinegar and season with salt and pepper. Let cool completely, about 30 minutes.
On a floured work surface, roll out the other disk of dough to a 12-inch round. Scrape the filling into the cooled pie shell and cover with the top crust. Trim the overhang to 1 inch and crimp the edge decoratively, sealing it to the bottom crust. Brush with the egg wash and cut slits in the top to vent steam. Bake the potpie for 40 minutes, until the crust is browned and the filling is bubbling. Let stand for 10 minutes before serving.
Schmaltz, rendered chicken fat, can be found in the freezer or refrigerated section of the grocery store or at your local butcher.