Preheat the Oven & Prepare Pans: Preheat the oven to 350°F (175°C). Grease and flour two 9-inch round cake pans or line them with parchment paper.
Melt the Chocolate: In a heatproof bowl, melt the chopped unsweetened chocolate over a double boiler or in the microwave in 20-second intervals, stirring until smooth. Set aside to cool slightly.
Mix Dry Ingredients: In a large bowl, whisk together the sifted cake flour, granulated sugar, baking powder, baking soda, and salt.
Cream Butter & Sugars: In a separate large bowl, beat together the softened butter, light brown sugar, and vanilla extract until light and fluffy.
Add Eggs & Milk: Add the eggs, one at a time, beating well after each addition. Gradually add the whole milk and mix until smooth.
Incorporate Chocolate: Mix in the melted chocolate until well combined.
Combine Wet & Dry Ingredients: Gradually add the dry ingredients to the wet ingredients in three parts, alternating with the sour cream. Start and end with the dry ingredients. Mix just until combined.
Bake the Cake: Divide the batter evenly between the two prepared pans. Smooth the tops and bake for 30-35 minutes, or until a toothpick inserted into the center comes out clean. Allow the cakes to cool in the pans for 10 minutes before transferring them to a wire rack to cool completely.