Make the cakes: Heat the oven to 350°. Line two 9-inch round cake pans with parchment paper circles, and spray with nonstick baking spray. In a large bowl, whisk the flour with the baking powder and salt. In the bowl of a stand mixer fitted with a paddle, cream the sugar with the butter on medium speed until pale and fluffy, about 3 minutes. Add the eggs, one at a time, beating well after each addition, until smooth. Reduce the mixer speed to low and add the dry ingredients, followed immediately by the milk, vanilla, and 1⁄2 cup water, and mix until the batter just comes together. Using a rubber spatula, stir the batter again until smooth.
Pour 2⁄3 cup of the cake batter into each prepared cake pan and, using a small offset spatula, spread the cake batter evenly over the bottom of each pan. Bake the cakes, rotating the pans halfway through cooking, until lightly browned and cooked through, 12 to 14 minutes. Transfer the pans to a rack, let the cakes cool for 10 minutes, and then invert the cakes onto the rack and remove the parchment paper. Once the cake pans are cool, re-line with parchment paper and coat again with nonstick baking spray. Repeat baking with the remaining batter to make 10 thin cakes total and let them all cool completely.
Make the fudge icing and assemble the cake: In a large copper or enameled cast-iron pan (these pans retain heat well, which helps prevent the icing from seizing), combine the sugar with the milk and place over medium-low heat, stirring until the sugar dissolves. Add the chocolate and butter and stir until the chocolate melts. Increase the heat to medium and cook, stirring occasionally, until the icing is thick and glossy and has the consistency of hot fudge sauce, about 15 minutes. Remove the pan from the heat and stir in the vanilla.
Place 1 cake layer on a cake stand and, using a small offset spatula, spread with 1⁄4 cup of the warm icing. Repeat stacking and spreading with 8 more cake layers and 2 cups icing, leaving the last cake un-iced. Spread the remaining icing over the top and sides of the cake stack until smooth. (If the icing hardens in the pan while you're icing the cake, rewarm it over low heat until it loosens before continuing.) Let the cake stand until the icing sets before serving. Store the cake, wrapped in plastic wrap, at room temperature for up to 3 days or in the refrigerator for 1 week.