Preheat your oven to 350℉. Line the bottom of an 8 or 9-inch round cake pan with a circle of parchment.
In a small bowl, combine the flour, baking powder and salt. Stir to mix well and aerate.
In a large bowl and using an electric mixer on medium speed, cream the butter, scraping down the sides of the bowl as necessary, until the butter is lighter in color, about 45 seconds. Then add the sugar in a steady stream and scrape down the bowl again. Continue to mix on medium speed until the mixture is very light in color and texture, about 4 to 5 minutes.
Slowly pour in the eggs a little at a time, and beat very well for about 2 minutes.
Combine the vanilla and milk in a small bowl or measuring cup.
With the mixer on low, add the flour mixture in four doses, alternating with the milk in three doses. Scrape down the sides of the bowl in between additions.
Pour the batter into prepared cake pan, place in the oven, and bake for 25 to 30 minutes, or until a toothpick inserted in the center comes out clean.
Remove from the oven and set on a rack to cool for 10 minutes then turn out onto rack to finish cooling completely.
Frosting
Heat chocolate in a double bolier until melted. Let cool to room temperature. In a large bowl using an electric mixer on medium speed, cream the butter. Add the chocolate, powdered sugar, milk, and vanilla and beat on low speed until combined, about 1 minute. Stop the mixer and scrape down the sides of the bowl. Increase the speed to medium and beat for 2 minutes, then reduce the speed to low. Add more sugar and milk as needed.
If the frosting is dry, add more milk, 1 teaspoon at a time, until it is creamy but still holds peaks.