Combine the butter, shortening, brown sugar and granulated sugar in a large bowl and beat with a mixer on medium-high speed until fluffy, about 3 minutes. Add the cocoa powder, baking powder, vanilla and salt and beat until combined. Reduce the mixer speed to low and add 1 1/2 cups flour and 3 tablespoons water. Add the remaining 1 1/2 cups flour and 3 more tablespoons water and beat until the dough is smooth. (Add 2 to 3 more tablespoons water if the dough seems dry.)
Preheat the oven to 350 degrees F and line 2 baking sheets with parchment paper. Divide the dough in half. Pat each piece into a flat square on a lightly floured surface, then roll out until to 1/4 inch thick. Cut into 2-to-3-inch squares with a paring knife or cut out shapes with cookie cutters. Transfer the cookies to the prepared baking sheets. Poke several holes in each cookie with a skewer (to look like a classic ice cream sandwich cookie), then transfer to the freezer until hard, about 10 minutes.
Bake the cookies until set, about 10 minutes. Let cool slightly on the baking sheets, then transfer to a rack to cool completely. Sandwich with ice cream and fillings; freeze at least 45 minutes before serving. To store, wrap the ice cream sandwiches individually in plastic wrap and freeze up to 2 days.
Ice cream and filling combinations:
Black raspberry ice cream, raspberries and fudge sauce
Mint chip ice cream and marshmallow cream
Chocolate ice cream, dulce de leche and fudge sauce