Line a baking sheet with parchment paper and place in the freezer. Working quickly and with one pint at a time, turn each pint of ice cream on its side and cut off the cardboard container. Starting with the wide end, slice off three or four 1/2-inch-thick rounds from each pint. Cut out stars from the ice cream using a 3 1/4-inch star-shaped cookie cutter, transferring the cutouts to the baking sheet in the freezer as you go. Freeze until solid, about 2 hours. (Transfer the ice cream trimmings to a resealable container and reserve for another use.)
Position a rack in the upper third of the oven; preheat to 325 degrees F. Butter a rimmed 13-by-18-inch baking sheet and line with parchment paper; butter the parchment, dust with flour and tap out the excess.
Stir the vanilla and red gel food coloring into the milk in a liquid measuring cup. Whisk the flour, malted milk powder, salt, baking powder and baking soda in a medium bowl. Beat the butter, shortening, granulated sugar and brown sugar in a large bowl with a mixer on low speed until combined. Increase the mixer speed to medium high and beat until pale and fluffy, about 5 minutes. Add the egg yolks, one at a time, and beat, scraping down the bowl as needed, until just combined. Reduce the mixer speed to low and beat in the flour mixture, then gradually beat in the milk mixture. Increase the mixer speed to medium and beat until smooth.
Spread the batter in the prepared pan and sprinkle with nonpareils. Bake on the upper oven rack until just set and a toothpick inserted into the center comes out clean, 25 to 30 minutes. Let cool completely in the pan, then loosen the edges with a paring knife and turn out onto a cutting board; discard the parchment paper. Cut out 20 stars as close together as possible using the same star-shaped cookie cutter. Sandwich the ice cream between the cookies, pressing gently to adhere. Freeze until firm, 1 to 2 hours.