For the cookies, combine the cocoa powder, salt, baking powder and flour in a medium bowl. In the bowl of a stand mixer fitted with the paddle attachment, cream the butter and both sugars until pale, about 3-4 minutes. Beat in the egg and vanilla extract, until the mixture is light and fluffy. Pat the dough into a 1/2-inch-thick disk and cover with plastic wrap. Chill in the refrigerator for at least two hours. (The dough can be made up to 2 days in advance).

For the ice cream, line a 13-inch x 9-inch baking pan with two sheets of plastic wrap. Let the chocolate ice cream soften for 5-10 minutes. Transfer the ice cream to a large bowl and whisk until smooth like thick cake batter. Use an offset spatula to spread the ice cream in an even layer in the prepared baking pan. Let the ice cream freeze until firm, about 20-30 minutes before repeating the same process with the vanilla ice cream. Cover and freeze until solid, about 8 hours.

Preheat the oven to 350 degrees F. On a lightly floured work surface, roll out the dough until it is 1/8-inch thick. Use a 2-1/2 inch round cookie cutter to cut out 24 cookies, rerolling the scraps as needed. Transfer to a parchment paper-lined sheet pan, spacing them 1-inch apart. Prick the tops of the cookies with the tines of a fork. Bake for 12-15 minutes. Cool completely on the baking sheets.

To assemble the sandwiches, turn over half the cookies. Use the 2-1/2 inch round cookie cutter to carefully cut out the ice cream. (Save the extra ice cream for another use). Wipe off the cookie cutter each time to avoid smearing the ice cream layers. Use a spatula to place an ice cream disc on each of the turned over cookies. Place another cookie on top right side up. Work quickly to limit ice cream melting. Freeze 2 hours before covering with plastic wrap and freezing for an additional 8 hours.