Heat the oven to 350 degrees F. Grease a 13 by 17-inch sheet pan and line the bottom and sides with parchment or waxed paper.
Sift together the 3 cups sugar, flour, cocoa, baking powder, baking soda, and salt. Transfer to a mixer fitted with a whisk attachment (or use a hand mixer) and blend briefly. Whisk together the eggs, milk, oil, and vanilla extract in a medium bowl and add to the dry ingredients and mix at low speed for 5 minutes. Gradually add the hot water, mixing at low speed just until combined. The batter will be quite thin. Pour the batter into the pan. Bake until a tester inserted in the center of the cake comes out clean (a few crumbs are okay) and the center feels firm to the touch, 25 to 30 minutes. Let cool in the pan. Chill, covered, until ready to build the cake. Make room in your freezer for a serving platter that you'll build the cake on.
Place a piece of parchment over the entire cake and flip it out onto another sheet pan. Peel off the original parchment and cut the cake in half so you have 2 pieces 13 by 8 1/2 inches. Place a piece of cake on a rectangular serving platter and schmear the ice cream on it all the way to the edges. Smooth it out with an offset spatula and place back in the freezer.
After it has chilled 15 minutes or so place the second layer of cake on top, cover with parchment paper and press down with a sheet pan gently to stick the top layer. Return to the freezer while you make the chocolate topping.
In a small saucepan, heat the 1 cup of cream and once it comes to a boil pour it over the chopped semisweet chocolate and whisk to make a smooth ganache. Do the same with the remaining 1/2 cup of cream and the white chocolate, adding the mint extract and green coloring to make a light mint green color then place it in small pastry bag with a small plain tip.
Remove the cake from the freezer and working quickly spread the chocolate ganache over the surface of the cake smoothing it with an offset spatula.
Drizzle lines of the green ganache over the entire surface of the cake at diagonal. Take a toothpick and run it back and forth in the opposite direction to draw it through the ganache and make decorative pattern.
Return to the freezer until ready to serve. Trim off the edges to even off the cake then cut into rectangle pieces.