Preheat oven to 300 degrees. Pulse flour, sugar, cocoa powder, and salt in a food processor to combine. Add butter and vanilla, and process until mixture comes together. Shape the dough into a ball.
Roll out dough on a lightly floured surface to 1/4 inch thick; cut out shapes with a 3 1/2-by-2 1/2-inch diamond-shape cookie cutter. Space 1 1/2 inches apart on baking sheets lined with parchment paper. Bake cookies until firm to the touch, 20 to 25
minutes. Let cookies cool completely on wire racks.
Melt white chocolate, stirring, in a heatproof bowl set over a pan of simmering water. Drizzle chocolate over tops of cookies. Let set before serving, about 30 minutes.
Cookies can be stored in a single layer in an airtight container at room temperature up to 3 days.