To make the cake, heat the oven to 350°F with a rack in the middle position (if your oven cannot comfortably fit 3 cake pans on a single rack, position the racks in the upper- and lower-middle positions).

Mist three 9-inch round cake pans with cooking spray. Line the bottoms with rounds of kitchen parchment, then mist the parchment. Dust the pans with flour, then knock out the excess.

In a medium bowl, combine the unsweetened chocolate and vanilla. Pour the hot coffee over the mixture, then let stand for 2 to 3 minutes to soften the chocolate.

Whisk until completely smooth, then cool until barely warm to the touch.

In a large bowl, whisk the eggs and oil until well combined.

Whisk in the sour cream, then add the cooled melted chocolate mixture, then whisk until homogeneous.

In a stand mixer with the paddle attachment, mix the flour, sugar, baking soda and salt on low until well combined and aerated, 2 to 3 minutes.

With the mixer running on low, add the chocolate-sour cream mixture to the dry ingredients in 3 additions, beating for 45 to 60 seconds after the first 2 additions and scraping the bowl as needed. After the final addition, increase to medium and beat until well combined, about 1 minute.

Fold the batter with the spatula, scraping along the bottom and sides of the bowl to ensure no pockets of flour or chocolate mixture remain; the batter will be thin.

Divide the batter evenly among the prepared pans. Rap each pan against the counter to remove any air bubbles. Place the pans in the oven, spacing them evenly on the rack (if baking on 2 racks, place 2 pans on the upper rack and 1 on the lower).

Bake until a toothpick inserted into the center of the cakes comes out clean, 35 to 40 minutes (if using 2 oven racks, 25 minutes into baking, quickly move the 2 pans on the upper rack to the lower rack and the pan on the lower rack to the upper rack, then continue to bake).

Let the cakes cool in the pans on wire racks for about 15 minutes.

Run a paring knife around the inside edge of each pan to loosen the cakes. One at a time, invert the cakes onto the racks, lift off the pans and peel off and discard the parchment.

Carefully turn the cakes upright and cool to room temperature.

To make the frosting, in a medium saucepan over medium, bring about an inch of water to a simmer. Put the semisweet chocolate into a heatproof medium bowl and set the bowl on top of the saucepan; make sure the bottom does not touch the water.

Stir occasionally until the chocolate is completely melted. Remove the bowl from the pan and set aside until cool to the touch, stirring occasionally.

In a stand mixer with the paddle attachment, beat the butter on medium until smooth and creamy.

With the mixer running on low, add the cooled chocolate and mix until fully incorporated, 2 to 3 minutes, scraping down the bowl as needed.

Add the milk and vanilla; beat on low until combined, about 30 seconds.

With the mixer running on low, gradually add the powdered sugar, then continue beating until fully incorporated and the frosting is smooth, about 1 minute, scraping the bowl as needed. Increase to medium and beat for 1 minute to slightly aerate.

To assemble the cake, using a serrated knife and a gentle sawing motion, carefully slice off the domed surface from each of the 3 cake layers, creating level tops.

Set 1 layer cut side down on a cake plate. (If you like, to keep the plate clean as you frost the cake, cut 5 kitchen parchment strips, each measuring roughly 6-by-3 inches, and tuck them under the edges of the bottom cake layer so they cover the plate).

Scoop about ¾ cup frosting onto the center of the cake and, using an icing spatula, spread in an even layer to the edges of the cake. Place a second cake layer cut side down on top, aligning it with the bottom layer, then frost it in the same way.

Place the final cake layer cut side down on top. Spread the remaining frosting onto the top and sides of the cake. (Remove and discard the parchment strips, if used.)