Mix the flour, sugar, salt, and baking powder in bowl of a food processor. Add the shortening and process until well incorporated into the flour.
Add the butter pieces and pulse 4 to 6 times until pea-sized nuggets of butter remain.
Add the water and vinegar and pulse until the dough is just holding together.
Wrap in plastic, flatten into a 5-inch disc, and chill in the refrigerator at least a few hours, preferably overnight.
Roll the pie dough into a 13-inch, 1/8-inch-thick circle, and fit the dough into a standard 9-inch pie plate. Trim the edges and freeze the crust for at least 20 minutes prior to blind baking. (While the crust is freezing, preheat the oven to 350°F or 176°C.)
Line the frozen pie crust with a sheet of aluminum foil or parchment paper and fill with pie weights. Bake the crust for about 50 minutes, or until the crust is just starting to turn a light golden brown in spots (you will have to check under the foil to see this). The goal is for the crust to be almost fully baked at this point.
Remove the foil and pie weights. Cool the crust to room temperature on a wire cooling rack.
Heat a water bath to 150°F (65.6°C).
Whisk the eggs in a bowl. Whisk in the brown sugar followed by the cream, vanilla, espresso powder, and salt.
Place the chocolate in a 1-gallon zipper-top (e.g., Ziploc) bag. Add the egg mixture and seal the bag by simple air displacement. The temperature of the custard at this point is essentially refrigerator temperature, 40°F to 50°F (4.4°C to 10°C).
Place the bag in the water bath for 15 minutes being sure to clip the top of the bag so that the zipper of the bag stays above the water level.
After 15 minutes, the chocolate will be melted and the temperature of the mixture at will be about 105°F (40°C).
You can agitate the bag by gently squishing it with your hands a bit to get everything mostly homogeneous.
Preheat the oven to 275°F (121°C).
Place the bag back in the water bath until it gets to just about 150°F (65.6°C). This should take about another 15 minutes. Agitate the bag again to finish mixing the contents (the bag will be much warmer, so be careful!).
Pour the custard into the cooled pie crust. Carefully transfer the pie to the oven (the pie will be quite full). Bake for 45 to 50 minutes, until the center of the pie measures 180°F (82.2°C) on an instant-read thermometer.
Cool the pie completely on a wire cooling rack.
Once the bottom of the pie plate is no longer warm, refrigerate the pie until it is cold throughout before serving.