Preheat oven to 350°. In a food processor, pulse cookies to a fine crumb. Add 2 tsp. sugar and the butter and pulse to combine. Gently press mixture over bottom and up sides (but not onto rim) of a 9-in. pie pan, working it evenly into the inside edge. Bake crust until just set (still soft in center when lightly touched; it will firm up as it cools), 8 to 10 minutes. Let cool completely on a rack.
Heat milk to simmering in a medium saucepan over medium heat. In a large bowl, whisk together egg yolks, 3/4 cup sugar, and the flour and cocoa powder. Whisking constantly, stream in warm milk. Transfer mixture to saucepan used for milk and bring to a boil over medium heat, stirring constantly. Reduce heat to medium-low and continue to cook, stirring constantly, until a finger drawn across mixture on back of a spoon leaves a trail, 1 to 2 minutes.
Strain mixture through a fine-mesh strainer into a bowl. Add chopped chocolate and the vanilla and stir until melted. Pour filling into cookie crust and chill uncovered until filling is firm and set, about 4 hours.
In a large bowl using a mixer on medium-high speed, beat cream and the remaining 1/4 cup sugar until soft peaks form. Mound over pie and sprinkle with shaved chocolate.
*To shave, set chocolate on a piece of waxed paper and microwave until slightly warm, 15 to 20 seconds; then shave with a vegetable peeler.
Make ahead: Through step 3, up to 1 day, chilled airtight; through step 4, up to 2 hours, chilled.