In a food processor, pulse the flour, cocoa, and salt. Add the butter and process until sandy, about 3 minutes. Add the confectioners' sugar, milk, and vanilla extract and process until a firm dough forms. Transfer the dough to 2 sheets of plastic wrap and form into two 7-inch logs, about 1 1/2 inches thick. Wrap in plastic and refrigerate until very firm, at least 4 hours or overnight.

Preheat the oven to 350 degrees and position racks in the upper and lower thirds. Line 2 large baking sheets with parchment paper. Beat the egg white in a small bowl. Sprinkle the granulated sugar in a 7-inch square on a sheet of wax paper. Brush the logs with the egg white and roll in the sugar, pressing to help it adhere. Cut the logs into 1/4-inch slices and transfer to the baking sheets, about 1/2 inch apart. Bake the cookies for about 20 minutes, until just firm to the touch; shift the sheets from top to bottom and front to back halfway through for even baking. Transfer the baking sheets to racks to cool.

The cookies can be stored in an airtight container for up to 2 weeks.