1. Whisk together flour, baking powder, and salt in a medium bowl. Beat together butter and sugar in a large bowl with an electric mixer on medium speed until light and fluffy, about 2 minutes. Add eggs and the vanilla; beat until combined well. Reduce speed to low and gradually add flour mixture, beating until just combined.
2. Transfer to a lightly floured surface and knead until dough holds together. Flatten into a disk, wrap well in plastic wrap and chill until firm, about 2 hours.
3. Preheat oven to 350 degree F. Arrange racks in upper and lower thirds of oven. Lightly grease 2 large cookie sheets.
4. Unwrap disk and roll out dough on a lightly floured surface with a floured rolling pin to about 13 inches in diameter and 1/4 inch thick. Cut dough into a 12-inch square, then cut square into 24 (2- by 3-inch) rectangles. Roll out remaining dough scraps to make 4 more cookies. Using heart, spade, diamond, and club cookie cutters, cut out shapes in the upper left-hand corner, middle, and lower right-hand corner of rectangles. Transfer to prepared cookie sheets, 1 inch apart. Lightly beat remaining egg and brush over cookies. Sprinkle with decorating sugar.
5. Bake until pale golden, switching positions halfway through, 10 to 12 minutes. Using a metal spatula, transfer cookies to racks and cool completely.
6. Remove one pint of ice cream from freezer to soften 15 minutes before assembling sandwiches. Place 2 cookies, bottom sides up, on work surface and top 1 cookie with about 1/2 cup ice cream; cover with another cookie, pressing down gently. Using your finger or a knife, even out the sides, being careful not to get ice cream on the edges of the cookies. Wrap tightly in plastic wrap and immediately place cookie sandwich in freezer. Repeat to make 6 more sandwiches, freezing as each is assembled. If ice cream gets too soft, freeze a few minutes to harden slightly. Repeat process with second pint of ice cream and cookies to make the remaining 7 sandwiches.
7. Allow ice-cream sandwiches to set in the freezer at least 4 hours before serving. Makes 14 sandwiches or 28 cookies.