Set the oven to 325°F (165°C). Line three 8-inch round cake pans with parchment circles and lightly grease the sides; dust with cocoa or flour and tap out the excess. For flatter layers, wrap the pans with damp cake strips if you have them.
Bloom the chocolate: In a large heatproof measuring cup, whisk the hot brewed coffee, cocoa powder, and instant espresso until smooth and glossy. Whisk in the cinnamon and 2 teaspoons vanilla; let cool 5 minutes.
In a large bowl, whisk together the flour, baking powder, baking soda, and salt until evenly combined.
In the bowl of a stand mixer fitted with the paddle (or using a hand mixer), beat the oil with the granulated and brown sugars on medium speed until sandy and glossy, about 1–2 minutes. Add the eggs one at a time, mixing for 30 seconds after each addition and scraping the bowl as needed.
On low speed, add half of the dry ingredients, then stream in about half of the coffee-cocoa mixture. Repeat with the remaining dry ingredients and coffee mixture, mixing just until combined. Fold in the sour cream by hand until the batter is smooth. Let the batter rest 5 minutes to settle air bubbles.
Divide the batter evenly among the prepared pans (about 1 2/3 cups per pan). Tap pans firmly on the counter to release large bubbles.
Bake for 26–30 minutes, rotating the pans halfway through, until the centers spring back lightly and a toothpick comes out with a few moist crumbs. Cool in pans 10 minutes, then invert onto wire racks and cool completely.
Make the mocha buttercream: Warm 3 tablespoons of the heavy cream until steamy and stir in the espresso powder to dissolve. Chill this espresso cream until cold to the touch (5–10 minutes).
Beat the cold, cubed butter on medium-low until smooth and slightly lightened, 3–4 minutes. Add about half of the powdered sugar and mix on low until incorporated. Pour in the chilled espresso cream, the remaining 3 tablespoons heavy cream, vanilla, and salt; beat on medium-high until fluffy, 1–2 minutes. Mix in the cooled melted dark chocolate. Add the remaining powdered sugar gradually, beating until silky and spreadable, 2–3 minutes more. Adjust with a splash of cream if needed.
Assemble: Level the cake layers if necessary. Place the first layer on a board. If using, lightly brush the top with cooled brewed coffee. Spread about 3/4 cup frosting evenly to the edges. Repeat with the second layer.
Top with the final layer, cut side down. Apply a thin crumb coat of frosting around the cake and chill for 15 minutes to set. Finish with a thicker coat, smoothing the sides or creating swoops as you like. Garnish with cocoa nibs or chocolate-covered espresso beans.
Slice with a warm knife for clean cuts. Serve at cool room temperature.