Preheat oven to 350°. Line an 8-in.-square pan with parchment paper to create a 1-in. overhang on two sides.
In a medium bowl, whisk together flour, cocoa powder, baking powder, and salt. In another medium bowl, whisk together butter, sugar, eggs, espresso powder, and vanilla. Add flour mixture and whisk until just blended. Pour batter into prepared pan and spread level.
Bake until a toothpick inserted in center comes out with a few moist crumbs, 28 to 35 minutes. Cool completely on a rack, then chill, covered, until firm in center, preferably overnight but at least 4 hours.
Lift brownie out. Mark center of sides with toothpicks as a guide. Sliding in a long serrated knife at corners first, split brownie in half horizontally. Put bottom layer back in pan and spread ice cream on top, using an offset metal spatula or flexible plastic scraper. Replace brownie top and gently press down.
Cover and freeze until ice cream is firm, at least 3 hours. Lift out brownie and cut into 2-in. squares. If you like, wrap each in a strip of parchment paper (to keep hands clean) and serve immediately.
Make ahead: Frozen airtight, 1 week.