For the cake: Preheat oven to 350°F. Prepare 2 9-inch cake rounds with butter and flour. Tap out excess flour.
Place chocolate in a stainless steel bowl. Place bowl over a saucepan of simmering water (do not allow bottom of bowl to touch water). Stir chocolate frequently with a whisk until completely melted. Remove from heat. You can also melt chocolate in the microwave in a microwave-safe bowl at 10-second intervals, stirring occasionally, until chocolate is melted. Reserve.
Sift together flour, baking soda, and salt. Reserve.
Using an electric mixer on medium-high speed, cream butter until soft – approximately 2 minutes. Add brown sugar and eggs. Beat with mixer at high speed until light and fluffy – approximately 5 minutes. Add vanilla extract and melted chocolate and mix until blended.
Stir in dry ingredients alternately with sour cream, mixing until batter is smooth. Stir in 1 cup boiling water (batter will be thin). Pour batter into prepared cake rounds. Bake until center springs back when lightly pressed with fingertip – approximately 35 minutes.
Cool cake in pans on rack for 10 minutes. Loosen around edges with a small knife. Remove cake from pans and cool completely on rack to room temperature.
For the frosting: Using an electric mixer on medium-high speed, cream butter until light and fluffy – approximately 2 minutes. Mix together milk and vanilla extract. Mix together cocoa and powdered sugar. Alternately add wet and dry ingredients to butter, blending on medium speed until well combined.
Place bottom layer cake on cake plate. Frost using offset spatula. Add top layer, frost top and sides.