Preheat oven to 350F. Line a 10 1/2 by 15 1/2 inch jellyroll pan with parchment paper or aluminum foil with the dull side up, pressing it along the edge and leaving an overhang on all sides. Butter the paper. Sift together the flour, cocoa, baking powder and salt.
In an electric mixer, beat the butter at moderately high speed until light and creamy. Scrape down the sides of the bowl. With the mixer on, gradually add the sugar; scrape down the sides of the bowl. Add the egg and vanilla and beat until well blended. Reduce the speed to very low and add the flour mixture. Mix until just barely blended; do not overmix. Turn off the machine and finish mixing by hand with a large rubber spatula (the dough should ahve the consistency of modeling clay; if too dry, add a few drops of cold water).
Transfer the dough to the prepared pan. Unsing your fingers, press the dough over the bottom of the pan to an even thickness. With a toothpick, make holes in the dough at 1/2 in intervals.
Bake until the cookie is set and the surface looks slightly dry, 8 to 9 minutes. Do not overbake, or it will be brittle. Transfer the pan to a wire rack. Cut the cookie in half crosswise while it is still hot, then let cool completely.
Place the ice cream in the refrigerator to soften until spreadable, about 10 minutes. Holding the edges of the parchment paper, lift off the cookie and place it on a work surface. Place a clean sheet of foil in the baking pan. With two spatulas, carefully transfer one of the cookie halves to the foil. Gently spread the ice cream over this half of the cookie in an even layer 3/4 inch thick, leaving a narrow border on all sides. With two spatulas, carefully set the second cookie on top and press very gently into place. Smooth the sides. Cover the sandwich wtih plastic wrap and freeze until the ice cream is hard, about 2 hours.
Remove the sandwich from the freezer and let stand for about 5 minutes to make it easier to cut. Dip a large chef's knife in hot water, wipe dry and cut the sandwich into 6 even rectangles. If you are not serving them immediately, wrap each sandwich individually in plastic wrap and store in the freezer for up to a week.