Adjust the racks to the center of the oven and preheat to 350 degrees F. Spray two 9-by-1-1/2-inch round cake pans with nonstick spray.
Add the granulated sugar and butter to the bowl of a stand mixer outfitted with the paddle attachment. Beat until creamy, just 2 minutes. Add the eggs one at a time to the mixture and beat until mixed together.
Whisk together the cake flour, all-purpose flour, baking powder and salt in a medium bowl. In a measuring cup, mix the milk and vanilla extract together. Turn the mixer back to medium speed and add the flour mixture and milk mixture alternately to create a smooth batter.
Evenly divide the cake batter between the prepared pans. Bake until the cakes start pulling away from the edges of the pan and a toothpick inserted into the center of the cakes comes out clean, 30 to 35 minutes. Cool the cakes for 10 minutes in their pans, then turn them out on a wire rack to cool completely before frosting.
Frost the cake by tucking parchment underneath the first cake layer and the cake plate. Spread the Chocolate Frosting evenly over the first layer, then place the second cake on top. Frost the top of the cake, then work your way over the sides. Remove the parchment and serve.
Add the chopped chocolate to a heatproof bowl and set over a lightly simmering pot of water (make sure the bottom of the pot is not touching the water). Stir until the chocolate is completely smooth and melted. Remove from the heat and cool until room temperature, about 15 minutes.
Add the confectioners' sugar to a medium bowl and whisk out the lumps.
Beat the butter in a stand mixer outfitted with the paddle attachment until creamy. Add the cooled melted chocolate. Gradually add the confectioners' sugar in 3 batches and beat until combined. Add the milk, vanilla and salt, and beat until the mixture is creamy, well mixed and spreadable.