Combine flour, cocoa powder, powdered sugar, and salt in a food processor. Pulse briefly to mix.
Add cold butter cubes and pulse until the mixture resembles coarse breadcrumbs.
Beat together the egg and vanilla extract. With the processor running, add the egg mixture through the feed tube.
Add ice water one tablespoon at a time until the dough just comes together.
Form the dough into a disk, wrap in plastic, and refrigerate for at least 1 hour or overnight.
Preheat oven to 350°F (175°C).
Roll out the chilled dough between two pieces of parchment paper to fit a 9-inch tart pan with removable bottom.
Carefully transfer the dough to the pan, pressing it into the corners and trimming excess.
Freeze the lined tart pan for 15 minutes.
Line the tart shell with parchment paper and fill with pie weights or dried beans.
Blind bake for 20 minutes, then remove weights and paper and bake for an additional 5-10 minutes until fully cooked.
Let cool completely before filling.