Begin by creaming together 1/2 cup of softened butter with 1 (16 ounce) package of brown sugar, beating well until the mixture is light and fluffy. Add 3 eggs one at a time, beating thoroughly after each addition to guarantee proper incorporation. Stir in 3 ounces of melted and cooled unsweetened chocolate, mixing until the chocolate is evenly distributed throughout the butter mixture.
In a separate bowl, combine 2 1/4 cups of sifted cake flour, 2 teaspoons of baking soda, and 1/2 teaspoon of salt. Gradually add this dry mixture to the creamed ingredients, alternating with 1 (8 ounce) carton of sour cream. Beat well after each addition to create a smooth batter. Once fully combined, add 1 cup of piping hot water and 1 1/2 teaspoons of vanilla, stirring well. The batter will be quite thin at this point, which is normal and necessary for achieving the cake's characteristic moist, fudgy texture.
Divide the batter evenly among 3 greased and floured 9-inch cake pans. Bake at 350°F for approximately 25 minutes, or until a toothpick inserted in the center comes out clean. Allow the cakes to cool in their pans for 10 minutes before turning them out onto wire racks to cool completely. This cooling period in the pans helps the delicate cakes set properly before handling.