Preheat oven to 350 degrees. Coat two 13- by-18-inch rimmed baking sheets with cooking spray, line with parchment, and spray parchment.
Whisk flour, baking powder, and salt in a medium bowl. Melt butter and chocolate in a large heatproof bowl set over a pot of simmering water, stirring until smooth. Remove from heat, add sugar, and stir until dissolved. Stir in eggs, 1 at a time, and then stir in vanilla. Stir in flour mixture until just incorporated. (Do not overmix.)
Divide batter between pans. Use an offset spatula to spread batter into thin, even layers. Bake, rotating pans once, until batter just begins to set, 8 to 10 minutes. (The surfaces will look dry, and testers inserted in the centers will come out wet.) Let cool completely in pans on wire racks.
To assemble, place the flat side of a large wire rack over 1 brownie, and invert rack, brownie, and pan together. Remove pan and parchment. Repeat with remaining brownie. Place a clean rimmed baking sheet over 1 brownie, and invert pan, brownie, and rack together. Remove rack. The brownie should be in the pan, top side up.
Spread ice cream in a smooth, even layer over brownie in pan. Align pan lengthwise with remaining brownie on rack, and use rack to quickly and carefully flip brownie onto ice cream. Remove rack. Press top brownie gently. Cover and freeze until firm, at least overnight (or up to 1 week).
Uncover, and cut into squares. Serve immediately (or cover and freeze ice cream sandwiches overnight).
If a rimmed baking sheet doesn't fit in your freezer, cut the brownies in half crosswise. Place one brownie in a 9-by-13-inch pan, cover with half of the ice cream, and top with another brownie. Cover and freeze. Repeat with remaining brownies and ice cream. Use your favorite brand of conventional peanut butter here; the flavor and texture of all-natural versions do not work well in this recipe.