Preheat the oven to 350 degrees F. Grease a 13x18 inch rimmed baking sheet and line the bottom with parchment paper or aluminum foil, leaving a 2 inch overhang on the two shorter sides.
In a mixer fitted with the paddle attachment, mix the butter and sugar on medium speed until light and fluffy, about 1 minute. Beat in the egg. Add the flour, cocoa powder, and baking powder. Mix until incorporated, about 1 minute more. Make sure to scrape the bowl to ensure ingredients are fully combined.
Using a mini offset spatula, spread the batter evenly in the prepared pan. Make sure to smooth the top. Bake for 8 minutes (start checking after 6-7 minutes). Remove from oven and let cool completely in the refrigerator.
After the cake has completely cooled, gently remove it from the pan and invert it onto a cutting board. Remove the parchment, cut the cake in half, and place one of the halves on a large piece of foil.
Remove the ice cream from the freezer and transfer to a mixer fitted with the paddle attachment. Start at low speed and continue mixing until it reaches the consistency of soft-serve ice cream. Using a mini offset spatula, spread the ice cream over the cake half that is on the foil. Slide the other cake half off the cutting board and onto the ice cream. Press down gently. Wrap the foil up and around the assembled dessert. Place in the freezer at least 3 hours or overnight.
When ice cream is fully set, remove the dessert from the freezer and trim the edges if necessary. Cut the rectangle into 8 equal pieces. Individually wrap each ice cream sandwich with parchment paper, wax paper, or aluminum foil and tape to seal. Store the sandwiches in an airtight container in the freezer for up to a week.