Preheat oven to 350 degrees F (175 degrees C). Grease and flour a 9x13-inch baking pan.
Melt semi-sweet chocolate and unsweetened chocolate in the top of a double boiler over simmering water, stirring frequently and scraping down the sides with a rubber spatula to avoid scorching. Remove double boiler from heat and cool chocolate to room temperature.
Beat eggs into cooled chocolate using an electric mixer until smooth.
Melt butter in a large saucepan over low heat; remove saucepan from heat. Stir sugar, chocolate mixture, and vanilla extract into butter until smooth. Beat flour, baking powder, and salt into butter-chocolate mixture until batter is just mixed; pour into the prepared baking pan.
Bake in the preheated oven until the top is dry and the edges have started to pull away from the sides of the pan, 15 to 20 minutes. Allow brownies to cool in pan.
Cut brownies into 24 squares. Slice each square horizontally and spread ice cream into the middle; place top brownie piece over ice cream. Wrap each ice cream sandwich with plastic wrap and freeze.
