In a large bowl, add 1¼ cup flour, ¼ cup + 2 Tbsp cocoa powder, ½ tsp salt and mix together until combined.
Cut the ½ cup butter into cubes and cut into the dry ingredients until butter is pea-sized and evenly coated.
Add in the wet ingredients: ¼ cup iced water, 2½ Tbsp beaten egg, ½ tsp vinegar, and mix together vigorously with a fork until just combined. Do not overmix. You should still see clumps of butter in the pie crust.
Divide crust into two portions. Wrap one in plastic wrap and put into the fridge for later. Roll the other crust out to ¼" and press into a 9" pie plate. Using a fork, poke holes in the bottom of the crust, then trim the edges.
If desired, lay a sheet of parchment paper over the crust and top with pie weights. Bake at 350°F for 20 minutes. Remove from oven and set aside.