Preparation

Make the cake

• Position racks in the upper and lower thirds of the oven and heat the oven to 350°F. Oil the bottom and sides of three 9-inch cake pans, line with parchment, and lightly oil the paper.
• Whisk the flour, cocoa powder, baking powder, and baking soda in a large bowl.
• In another large bowl, whisk the egg yolks with 1/2 cup of the sugar and the salt until the sugar begins to dissolve, about 2 minutes. Whisk in the oil, coffee, and vanilla, and then the dry ingredients, until just incorporated.
• In a stand mixer fitted with the whisk attachment, whisk the egg whites on medium-high speed until soft peaks form. With the mixer running, slowly add the remaining 1 cup of sugar, beating until very glossy with firm peaks, 3 to 4 minutes.
• Whisk one-third of the egg whites into the batter to lighten it; then, using a large spatula, fold in the remaining whites until completely incorporated. Divide the batter equally among the prepared pans, leveling with an offset spatula. (The batter will be about 1/2 inch deep.)
• Bake all three layers on the lower rack if they fit, or set two pans on the lower rack and the third on the upper rack, staggering them so that no pan is directly over another. Bake, rotating the pans and swapping positions halfway through, until the tops spring back when touched and a toothpick inserted in the center comes out with moist crumbs adhering, 16 to 18 minutes. Cool on racks for about 10 minutes. Run a thin knife around the edge of each pan and turn the layers out onto the rack; peel off the parchment and turn the layers right side up. Let cool completely.

Assemble the cake

• Line a 2-1/2-quart bowl, preferably one that’s 9 inches in diameter and about inches high, with plastic wrap; you may need to use two pieces. Cut a 6-inch circle from one of the cake layers and an 8-inch circle from another cake layer. (Plates make a good guides for cutting.) Place the 6-inch cake layer in the bottom of the lined bowl.
• Let one of the quarts of ice cream soften at room temperature for 5 minutes. Cut the container down the side to remove the ice cream. Cut the ice cream into large chunks and put them in a stand mixer fitted with the paddle attachment. Beat on low speed until the ice cream is spreadable, about 1 minute. Using an offset spatula, spread the ice cream evenly over the cake layer in the bowl. Top with the 8-inch cake layer, cover with plastic wrap, and put in the freezer.
• Soften and beat the remaining quart of ice cream the same way. Remove the bowl from the freezer and spread the ice cream in an even layer over the cake. Top with the final cake layer, trimming if necessary to fit; it’s OK if the layer is above the rim of the bowl. Cover with plastic wrap and freeze until completely firm, at least 4 hours and preferably overnight.

Make the sauce

• In a 1- to 2-quart saucepan, bring the heavy cream to a boil. Remove from the heat and add the chocolate, syrup, and rum. Let sit for 5 minutes. Whisk until completely smooth.

Make the meringue topping

• Cut a 10-inch circle out of clean heavy cardboard or use a pre-cut cardboard cake round.
• Take the cake out of the freezer and defrost for about 10 minutes. Uncover it and invert the bowl over the cardboard round. Remove the bowl (but leave the plastic wrap) and return the cake to the freezer until needed.
• Position a rack at the bottom of the oven and heat the oven to 500°F.
• In a stand mixer fitted with the whisk attachment, beat the egg whites and salt on medium-high speed until medium peaks form, 1 to 1-1/2 minutes. With the mixer running, slowly add the sugar, beating until medium-firm peaks form, about 2 minutes. Add the vanilla and continue beating until the whites are very glossy with firm peaks, 1 to 2 minutes.

• Put the cake (on its cardboard base) on a rack set over a baking sheet. Remove the plastic and spread the outside of the cake with the meringue, covering it completely right down to the cardboard base to seal in the cake. Using the back of a spoon, create swirls in the meringue and then use a fork to pull it into decorative spikes.
• Bake until the tips of the meringue are browned, 2 to 5 minutes. (Keep an eye on it through the oven window, as opening the oven door will inhibit browning. Alternatively, use a kitchen torch to brown the meringue.
• Transfer to a serving platter. To slice, use a large serrated knife, dipping it in hot water and wiping dry between each cut. Serve with the chocolate sauce. Refreeze any leftover cake covered in plastic wrap.

Make Ahead Tips

The cakes can be made 1 day ahead and kept wrapped in plastic at room temperature. Well-wrapped layers can also be frozen for up to 2 months.

The sauce can be made and kept at room temperature for up to 3 hours; for longer storage, transfer it to a container, cool, cover, and refrigerate for up to 1 week. Reheat gently before serving.

The entire dessert, minus the meringue, can be assembled and kept frozen for several days before covering with meringue, baking, and serving.