If the ice cream isn't already softened, place container of chocolate ice cream at room temperature for about 10 minutes while you prep the cake.
Slice the sponge in half horizontally to make two layers. On a flat tray or plate that will fit into your freezer, place a sheet of parchment or foil. Place one cake layer as the base of the tiramisu cake.
Place a 9"-wide sheet of parchment paper or foil in 9" square cake pan, allowing edges to hang over sides of pan. You will be using the parchment paper to take molded ice cream out of the pan.
Spread softened chocolate ice cream in paper/foil-lined cake pan, making sure the top is smooth and flat. Place pan in freezer to let ice cream harden. While chocolate ice cream layer is hardening, let coffee ice cream soften.
When the chocolate ice cream layer has hardened to somewhat hold its shape, pull the ice cream layer out of the cake pan with the edges of the parchment and carefully place on top of the base cake layer. Press down gently on the ice cream with a wide spatula to anchor the ice cream into the cake. Place in freezer to keep the ice cream from melting.
You can use the same parchment liner or tear/cut a new sheet, and place back into cake pan. Spread softened coffee ice cream in cake pan, again, making sure top is smooth and flat. Place in freezer for coffee ice cream layer to harden.
When the coffee ice cream layer is sufficiently hardened, remove from freezer along with the cake+chocolate ice cream layers. Place second cake layer on top of the chocolate ice cream layer and press it down gently into the ice cream layer. Remove the coffee ice cream layer from the pan and carefully place on top of the second cake layer. Press down gently. Place entire cake back into freezer to allow coffee ice cream to harden, about 15 minutes.
While cake is hardening, make Mascarpone Cream.
Remove cake from freezer. With 2 very large spatulas, carefully move cake to serving platter.
Spread top of cake with Mascarpone Cream. Dust with cocoa powder. Top with chocolate curls.
Serve immediately. Freeze the leftovers.