Make the butterscotch icing

Cup of Yum
1. Combine the sugars, corn syrup, butter, cream, and salt in a large, heavy-bottomed saucepan to which you've attached a candy thermometer. Place it over medium-high heat. Cook, stirring continuously, until the butterscotch mixture comes to a boil. Have another pot of a similar size ready and on the side, off the heat. When the butterscotch mixture begins to boil, stir it frequently, taking care not to scrape the side of the pan. Cook at a full boil until the mixture reaches 242°F (120°C). This could take as long as 30 minutes. Be patient.
2. Immediately pour the butterscotch into the empty pot. (Do not scrape the sides or bottom of the saucepan with the spoon, as this could cause the icing to crystallize.) Immediately and carefully add the vanilla and rum to the icing, pouring from a good height above the pot. (If you pour from too close to the pot, the icing will steam and potentially burn you.) Stir to combine. Let the icing cool for 20 minutes, at which point it should be thick yet pourable.

Make the cake

1. Cut ten 8-inch circles of parchment or waxed paper. The baking will progress more quickly if you have at least three 8-inch cake pans. Preheat the oven to 350°F (175°C).
2. Combine the flour, baking soda, baking powder, salt, and nutmeg in a medium bowl and whisk lightly to combine. Using an electric mixer fitted with the paddle attachment, beat the butter and sugar until light and fluffy Add the eggs, 1 at a time, beating after each addition. Reduce the speed to low and add half of the flour mixture. Blend well. Add the vanilla, rum, and milk and mix just until the liquid is incorporated. Add the remaining flour mixture and beat until smooth, up to 4 minutes.
3. Working with 3 cake pans at a time, pour a heaping 1/2 cup of batter into each pan. Use a spatula to evenly spread the batter in the bottom of the pans. They will be very skinny cake layers. Bake for 15 to 18 minutes, or until the cakes spring back when lightly touched. Cool the layers in the pans slightly, then invert the cakes onto a wire rack to cool completely. Wipe the pans clean and continue to fill and bake with the remaining batter. You will need 9 layers for this cake.
4. When all the cakes have been baked, line the inside of a clean, dry cake pan with plastic wrap. (If the plastic wrap sticks out above the rim of the cake pan, fold it over the outside of the pan.) Peel the paper from the top of one cake layer and invert the layer so it's upside down in the pan. (Or really, right side up, given that it was upside down on the wire rack.)
5. Ladle about 1/4 cup of the warm butterscotch icing over the cake. (If the icing is no longer warm, rewarm it gently over low heat.) Top with another cake layer, paper removed. Continue ladling the filling evenly over each layer. The cake will grow higher than the cake pan as you fill; the cake pan is there just for a little foundational stability. Don't worry if the edges of the cakes are a bit ragged; you'll trim them before icing the sides of the cake. Just make sure the cake layers are evenly stacked, as it's difficult to move a layer once it's been placed on the icing. If at any point the butterscotch icing becomes too thick to pour easily, place it over low, low heat. Do not top the last cake layer with butterscotch. Refrigerate the cake until chilled completely through, at least 1 hour.
6. Place an 8-inch circle of parchment on top of the cake. Carefully invert the entire stack of layers onto a wire cooling rack. Remove the pan and the plastic wrap. Hold a long, sharp knife lengthwise alongside the cake and trim about 1/4 inch evenly all around, making sure you are not tilting the blade. The cake should be uniformly shaped, not wider at the bottom than the top.
7. Gently reheat the remaining butterscotch icing and pour it over the top and sides of the cake. Let stand until the coating is firm. Slide a metal spatula under the cake and transfer it to a serving platter. This cake is best if made a day ahead, although it keeps beautifully for up to 5 days. Let it come to room temperature before slicing and serving.