a 3 1/2-inch round cookie cutter; a 3-inch round cookie cutter
For the ice cream: Line a 9-by-13-inch baking dish with plastic wrap. Spoon the ice cream into a large bowl, then add the preserves. Mix the preserves into the ice cream, leaving it somewhat swirled if possible. Spread the ice cream evenly in the prepared baking dish. Cover with plastic wrap and freeze until very firm, at least 3 hours.
For the cookies: Meanwhile, preheat the oven to 350 degrees F. Line 2 baking sheets with parchment paper.
Whisk together the flour, cocoa powder, baking powder and salt in a bowl.
Beat together the sugar and butter in a separate large bowl with an electric mixer until light and fluffy, about 3 minutes. Add the vanilla and egg and beat to combine. Add the flour mixture a little at a time, beating until incorporated; knead with your hands a few times to mix in the remaining flour and bring everything together.
Turn the dough out onto a lightly floured piece of parchment. With a floured rolling pin, roll the dough out to a 1/4-inch-thick circle, about 12 inches in diameter. Cut out as many 3 1/2-inch rounds as possible. Gather the scraps and cut out more rounds. Repeat until you have 12 rounds.
Arrange the cookies on the prepared baking sheets about 1/2 inch apart. Bake until just set on top, about 8 minutes. Cool completely on the baking sheets on a rack.
Assemble the cookies: Working quickly, remove the plastic wrap from the ice cream and use a 3-inch cookie cutter to cut out a round of ice cream (lift the plastic wrap from the bottom to help you lift out the round). Sandwich the ice cream between 2 cookies. Sprinkle on red and blue sprinkles along part of the sides and immediately transfer the ice cream sandwich to a plate in the freezer. Repeat with the remaining cookies to make 6 ice cream sandwiches. (The ice cream scraps can be gathered and kept frozen in an airtight container). Freeze the sandwiches until firm enough to serve, 3 to 4 hours.