Preheat your oven to 350°F (175°C). Grease and flour three 8x2 inch round cake pans, and for extra ease, line the pans with parchment paper.
In a microwave-safe bowl, combine the milk and dark chocolate. Microwave for 20 seconds, stir, and repeat for another 10 seconds. Let sit briefly before stirring again to melt completely. If needed, microwave in additional 5-10 second bursts. Stir until the mixture is smooth and glossy, then let it cool slightly.
In a separate bowl, whisk together the flour, cocoa, baking soda, baking powder, and salt for 30 seconds. Set aside.
In another bowl, combine the melted chocolate mixture with the sour cream and set it aside.
Using a mixer, beat the softened butter until smooth. Gradually add the sugar and continue beating for 4-5 minutes, until light and fluffy. Add the vanilla extract and mix to combine.
Add the eggs one by one, ensuring each yolk is fully incorporated before adding the next.
Gradually add the dry ingredients to the butter mixture, alternating with the chocolate-sour cream mixture, beginning and ending with the dry ingredients. Mix until just combined, scraping down the sides as needed.
With the mixer on low speed, slowly add the hot coffee and mix for about 30 seconds until fully incorporated.
Divide the batter evenly among the prepared pans and bake for 22-25 minutes, or until a toothpick inserted into the center comes out clean or with a few moist crumbs. Let the cakes cool in the pans for 5-10 minutes before turning them out onto a wire rack to cool completely.