Preheat oven to 350°. Beat butter and sugars together in a medium bowl with a hand or stand mixer on high speed until light and fluffy, about 2 minutes. Add egg and vanilla and beat until combined.
In a medium bowl, sift together flour, cocoa, baking powder, baking soda, and salt. With mixer on medium speed, add dry ingredients to creamed mixture in 2 batches, beating 15 to 20 seconds after each batch to incorporate and scraping sides of bowl as needed.
Lightly oil your hands and use them to roll dough into 16 walnut-size balls. Evenly space the balls on 2 parchment-lined cookie sheets and, using your palms, press each ball into a 2-in. disk.
Bake cookies until they have spread out and are cooked but still soft, about 10 minutes. Cool slightly, then transfer to a baking rack to cool completely.
To make ice cream sandwiches, spoon 1/4 cup ice cream onto flat side of 8 cookies. Top with remaining cookies, flat side down, pressing lightly. Chill sandwiches in freezer about 10 minutes, just long enough to chill ice cream but not enough to harden cookies. Eat immediately.