Chill bowl, lid, and steel blade of food processor, plus all crust ingredients, in freezer for 1 hour. Preheat oven to 350 degrees.
Place flour, cocoa powder, sugar, and salt in bowl of food processor and process for about 1 minute to combine. Add butter and pulse 2 or 3 times, until pea-size pieces form. Slowly add water through feed tube, pulsing 4 or 5 times to incorporate.
Place dough onto lightly floured work surface and shape into disk. Start rolling by lightly pressing with rolling pin and rolling from center out. Do this a few times, then pick up dough, rotate a quarter turn, and roll again. Continue to roll and rotate dough, flouring work surface and dough as needed, until dough forms 12-inch circle that is 1/8-inch thick.
Loosely fold dough into quarters and then unfold in center of 10-inch pie pan. Gently fit it into bottom and up sides of pan. Should have approximately ½-inch overhang around edges. Use paring knife to remove extra dough and then fold dough under to create clean edge around entire rim. Let crust rest for 10 minutes to prevent shrinkage.
Line crust with foil or parchment paper and fill with pie weights. Place crust on rimmed baking sheet and bake 15 minutes. Remove crust from oven and remove weights and foil or parchment. Prick bottom of crust several times with fork. Return to oven and bake for approximately 7 minutes, until golden and appears dry. Cool completely before adding filling. Reduce oven temperature to 300 degrees.
For the filling:
Place butter and chocolate in top of double boiler. Fill bottom of double boiler with water, insert top, and set double boiler over low heat; water should never be hotter than a simmer. Stir butter and chocolate with silicone spatula until melted, scraping down sides as necessary and being careful not to incorporate air. Remove from heat.
Combine sugar, cocoa powder, and salt in large bowl. Whisk in milk, then whisk in eggs and vanilla. Fold in chocolate and butter mixture. Pour filling into chocolate crust.
Place in oven and bake for 30 minutes. Rotate pie front to back and bake for 15 additional minutes, or until filling is set and no longer jiggles in center. Let pie cool to room temperature on baking rack, then refrigerate for at least 2 hours before slicing. Serve at room temperature. Tightly covered, pie will keep in refrigerator for up to 3 days.