Preheat the oven to 325 degrees F. Line the bottom of two 8-inch-square metal baking pans with parchment paper and lightly butter the paper. Sift the flour, baking powder, baking soda and salt into a bowl, then whisk to combine.
Melt the butter in a saucepan over medium heat. Add the cocoa powder and whisk until dissolved, about 1 minute. Transfer to a bowl and whisk in both sugars until combined. Add the coffee, buttermilk, egg and vanilla and whisk until smooth. Whisk in the flour mixture until just combined.
Divide the batter evenly between the prepared pans. Bake until a toothpick inserted into the center comes out with a few moist crumbs, 15 to 20 minutes. Cool slightly in the pans, then invert onto a rack to cool completely. Transfer the cakes to a baking sheet lined with parchment paper, cover with plastic wrap and freeze until firm, about 1 hour.
Remove the ice cream from the freezer to soften, 10 to 20 minutes. Clean and dry the baking pans, then line each with plastic wrap, leaving a 4-inch overhang on all sides. Scoop all of the ice cream into one of the pans, packing it to make an even 1-inch-thick layer; smooth with a small offset spatula or the back of a spoon. Fold the plastic wrap over the ice cream to cover it, then freeze until solid, at least 1 hour.
Put 1 frozen cake in the remaining plastic-lined pan. Unwrap the ice cream layer and invert it on top of the cake; remove the plastic wrap. Cover with the remaining cake, pressing gently. Fold the plastic wrap over the cake to cover it and freeze until firm, about 1 hour.
When ready to serve, unwrap the cake and cut into 9 squares. Cut each square in half to make 2 rectangles or triangles.