Preheat the oven to 375 degrees . Grease and flour an 8-inch round cake pan. Line a 12-cup muffin tin with paper liners.
Using an electric mixer, beat together the butter and sugar until creamy. Add the eggs, 2 teaspoons vanilla, the salt and baking powder and mix until incorporated. Stir in the flour and 11/4 cups milk until smooth. Pour the batter into the prepared cake pan to fill about two-thirds full. Spoon the remaining batter into the prepared muffin tin. Bake until a toothpick inserted in the center comes out clean, about 30 minutes for the cupcakes and about 45 minutes for the cake. Let the cake cool for 5 minutes, then invert onto a rack to cool completely.
When the cake is cool, mix the confectioners sugar with the remaining 1/4 cup milk and 1 teaspoon vanilla. Cut the cake in half horizontally and frost the half-layer with a third of the frosting. Place the unfrosted half-layer on top and frost with another third of the frosting. Use the remaining frosting for the cupcakes.