Preheat oven to 300 degrees. Pulse flour, sugar, cocoa, and salt in a food processor to combine. Add butter and vanilla; process until mixture comes together. Form into a ball.
Roll out dough 1/4 inch thick; cut out shapes with a 3 1/2-by-2 1/2-inch diamond-shaped cookie cutter. Place on parchment-lined-baking sheets, and bake until firm to the touch, 20 to 25 minutes. Let cookies cool completely.
Melt white chocolate, stirring, in a heatproof bowl set over a pan of simmering water. Drizzle over shortbread. Let set before serving, about 30 minutes.