To prepare the cake:

1. Preheat the oven to 350 degrees. Grease a 13-by-9-by-2- inch pan with nonstick cooking spray or softened butter, line with parchment paper, and grease again Line a 9-by-5-by-3 inch loaf pan with a piece of plastic wrap that hangs slightly over all four sides.
2. In the bowl of a stand mixer fitter with the paddle attachment, combine the flour, cocoa powder, brown sugar, granulated sugar, baking soda, and baking powder, and salt on low speed until well incorporated.
3. In a bowl, whisk together the yolks, vanilla, oil, and buttermilk until combined. With the stand mixer on medium-low speed, slowly pour the wet mixture into the dry ingredients and beat until incorporated. Scrape down the bowl with a rubber spatula as needed.
4. In the same bowl, combine the boiling water and espresso powder, add it to the batter in the mixer bowl and mix for 30 seconds on low speed until smooth.
5. Pour the batter into the larger, parchment paper-lined pan and bake for 14-16 minutes, rotating at the halfway point. The cake is ready when a toothpick inserted in the middle comes out with a moist crumb or two. Let cool completely and freeze for 30 minutes or up to 3 days, tightly wrapped.
6. Place the strawberry ice cream in the microwave on high power for 20 seconds, or soften it on the counter until it scoops easily but isn’t melted. Place the ice cream on the bottom of the prepared loaf pan. Drape a piece of plastic wrap over the ice cream and use your hands to press it down into a flat, even layer.
7. Cut the cake into three 4-inc wide pieces. Place 1 piece of the cake over the strawberry ice cream and press down. Return the other two pieces to the freezer.
8. Soften the chocolate ice cream as you did the strawberry. Using plastic wrap and your hands, spread it into an even flat layer over the cake. Place another piece of the cake over the chocolate ice cream and press down. Repeat with the vanilla ice cream and the last piece of cake The last piece will be above the edge of the pan.
9. Freeze until hard, at least 4 hours, preferable overnight, tightly wrapped in plastic wrap.

For the meringue:

1. Whisk the egg whites in the bowl of a stand mixer on medium-high speed until foamy. Slowly add the granulated sugar, 1 tablespoon at a time, and whisk on high speed until stiff, glossy peaks form, at least five minutes.
2. Remove the loaf pan from the freezer and lift out the cake using the plastic wrap sling. Dip the bottom of the pan briefly in hot water if it has trouble releasing. Invert the cake onto a heatproof serving platter (the bottom layer is now the top) and generously cover the cake in meringue using an offset spatula or butter knife. Using a kitchen torch, if you have one, gently brown the meringue, or place the cake under the broiler for 1 or 2 minutes, watching closely to make sure it does not burn.
3. Slice the cake with a long serrated knife and serve immediately. It is best the day it is made, but will keep in the freezer, lightly wrap in plastic wrap, for up to 3 days.