First, make the pie dough. In a large liquid measuring cup, whisk together the water and vinegar. Add the ice and whisk, Refrigerate while you prep the rest of the ingredients.
Cut the butter into 1-inch cubes and place them in a small bowl. Freeze while you prep the rest of the ingredients.
In the bowl of a stand mixer fitted with the paddle attachment, combine the flour, sugar, and salt. Beat on low until just combined, about 15 seconds. Add the butter all at once and beat on low until the mixture has the texture of coarse meal, with pea-sized pieces of butter throughout, about 3 minutes.
Remove the ice water mixture from the refrigerator. With the mixer on low, add 4 tablespoons of liquid from the ice water mixture. Beat on low for 2 to 3 minutes, or until the dough clumps around the paddle and/or sides of the bowl. If the dough seems too dry, add more liquid from the ice water mixture 1 teaspoon at a time.
Tip the dough out onto a lightly floured counter with the spatula. Quickly knead the dough into a rough ball. Wrap the dough tightly in plastic wrap and flatten into a small disc. Refrigerate for at least 1 hour.
Form the bottom crust. Roll out the disc of chilled dough on a lightly floured counter and fit it onto a 9-inch pie pan. Crimp the edge and use a fork to poke holes all over the bottom and sides of the dough. Cover loosely with plastic wrap and freeze overnight.
Prep your oven and pan. Position a rack in the center of the oven and preheat the oven to 350°F. Line a sheet pan with parchment paper and place the frozen crust, still in its plate, in the center of the pan. Cover the crust with foil, making sure the crimped edges are completely covered and that there are no gaps between the foil and the crust. Fill with pie weights and spread them out so they are more concentrated around the edges of the crust.
Prebake the crust. Bake for 35 minutes. While the crust is in the oven, make the egg wash: In a small bowl, whisk together the egg, water, and salt.
Remove the sheet pan from the oven, keeping the oven on. Carefully lift out the pie weights and foil. Use a pastry brush to coat the bottom and sides of the pie crust (but not the crimped border) with a thin layer of egg wash. Bake, uncovered, for an additional 5 minutes. Set aside on a wire rack to cool slightly while you make the chocolate chess pie filling.
Make the chocolate chess pie filling. Place the butter and dark chocolate in the top pan of a double boiler or in a heatproof bowl set over a heavy-bottomed sauce pan filled with a few inches of simmering water (be sure the bottom of the bowl does not touch the water). Cook over medium heat, using a heatproof rubber spatula to stir the mixture and scrape the sides of the bowl occasionally, until the butter and chocolate have melted and combined, about 10 minutes. Set the double boiler or bowl on a wire rack and let the chocolate mixture cool while you prep the other ingredients.
In a medium bowl, whisk together the sugar, eggs, cocoa powder, vanilla, and salt. Slowly add the chocolate mixture and whisk to combine. Pour the mixture into the pie shell.
Bake the assembled chocolate chess pie. Bake for 50 to 55 minutes, or until the edges are set and the center of the pie just slightly jiggles. Cool on a wire rack.
Serve and store. Serve slightly warm or at room temperature. The pie can be kept at room temperature, under a cake dome or a large bowl turned upside down, for up to 3 days.