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To make the cake: Preheat the oven to 350°F. Lightly grease two 8” round pans; make sure they’re at least 2” deep. Divide the batter evenly between the two pans and bake the cakes for 35 to 38 minutes, until they test done.
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Turn the cakes out of the pans onto a rack to cool completely while you continue with the recipe.
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To make the syrup: Combine the sugar and water in a microwave-safe measuring cup and heat for 90 seconds at high power; or heat the mixture in a small saucepan set over medium heat. Stir until the sugar dissolves, then add the rum. Set aside to cool.
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To prepare the cake: Cut a small vertical notch in the sides of the two cake layers. (Since you’ll be cutting each layer in half to make a total of four layers, the notches will help you line them back up when it's time to assemble the cake.)
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Cut the layers in half horizontally to make four thin layers. Brush all four layers with the soaking syrup.
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Put one of the layers cut side down on a serving plate or cake round. Arrange the others, cut side down, on a couple of baking sheets; you’ll be working with them shortly.
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To make the filling: Place all the ingredients except the vanilla in a 2-quart saucepan. Cook over medium heat, stirring occasionally, until the sugar melts.
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Once the mixture comes to a boil, don’t stir. Cook until it reaches 238°F on a digital thermometer.
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Remove the filling from the heat, add the vanilla, and pour 1/4 cup (57g) into a small bowl; set aside. Let the remaining filling sit in the saucepan without stirring for about 8 minutes, to cool just a bit.
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Stir the still-warm filling with a sturdy spoon until it leaves lines on the bottom of the pan and begins to thicken. As soon as that happens, immediately pour it over the cake layer on the serving plate, plus two of the three layers on the baking sheets. (You have about 30 seconds before the filling starts to set.) Pour about 1/4 cup (75g) onto each layer, then repeat in sequence until all the filling is used. It’s OK if it runs over the edges a bit.
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To make the frosting: Make the syrup for the Italian Buttercream as directed in step #1 of the buttercream recipe.
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While the sugar is boiling, add 1 tablespoon milk to the reserved 1/4 cup filling and heat it for 30 seconds in the microwave (or in a small saucepan). Stir until it’s pourable; keep warm.
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Prepare the meringue as directed in step #2 of the Italian Buttercream recipe.
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When the syrup is between 240°F and 250°F, remove it from the heat and add the warm filling/milk mixture to it.
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To finish the frosting, pour the resulting filling/syrup mixture into the meringue in the mixing bowl and proceed with the Italian Buttercream recipe as directed.
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To assemble the cake: Spread 3/4 cup (139g) of frosting over each of the three filled layers.
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Stack the filled/frosted layers on top of each other, using the notches you cut to help align the layers. Place the last (unfilled) layer on top, cut side down. You now have a four-layer cake awaiting its final frosting.
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Cover the top and sides of the cake with the frosting; an offset spatula is an especially helpful tool here. Reserve 1 cup (185g) of the frosting to decorate the top, if desired. Refrigerate the cake while you make the ganache.
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To make the ganache: Place the chopped chocolate in a heatproof bowl.
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Heat the heavy cream and corn syrup in a small saucepan or in the microwave until the cream just starts to bubble around the edges. Pour the hot mixture over the chocolate and let sit for 3 minutes for the chocolate to soften.
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Stir the ganache until it’s smooth and shiny. If it looks grainy or lumpy no matter how much you stir it, an immersion blender will smooth it out nicely.
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Remove the cake from the refrigerator and pour the ganache over the top, letting some of it run down the sides.
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Once the ganache has cooled and set, use the reserved frosting to decorate the top of the cake, if desired.
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Storage information: Store the cake, well wrapped, in the refrigerator for up to five days, making sure plastic wrap is pressed directly against any cut edges. Freeze for up to six weeks.
