Preheat the oven to 350°F (176°C). Butter and flour at least four 9-inch cake pans.
In the bowl of a stand mixer or in a large bowl with an electric mixer, beat the butter and sugar until smooth. Add the eggs, 1 at a time, beating just until blended after each addition.
In a small bowl, mix the vanilla with the milk. Add the flour to the egg mixture alternately with the milk mixture, beginning and ending with the flour.
Pour a very thin layer of batter—about 7 tablespoons—into each pan, shaking the pans to distribute the batter to the edges. Bake the layers for 11 to 13 minutes.
While the first cake layers bake, return the glaze to low heat as the glaze must be warm to spread on the cake layers.
When the cake layers are done, immediately remove the layers from the pans and, working one at a time, place the layers on a cake stand and immediately slather with some of the warm glaze.
Bake all of the remaining batter in this manner and continue to stack and glaze in this fashion. You should be able to get 12 layers from this recipe. Reserve the last of the glaze to dribble over the top of the cake.