Heat the oven to 340°F. Grease two 8-inch round cake pans, line the bases with parchment, and lightly flour the sides. Set aside.
Bloom the cocoa: In a heatproof bowl, whisk the 3/4 cup cocoa powder (for the cake) with the hot coffee and espresso powder until smooth and glossy. Let cool until just warm, about 5 minutes.
Grate or finely shred the peeled sweet potato using the large holes of a box grater or a food processor. Gently blot with a paper towel to remove excess surface moisture without compressing the shreds.
Whisk the dry mix: In a separate bowl combine flour, baking powder, baking soda, salt, and cinnamon; set aside.
Build the batter base: In a large bowl, whisk together the oil, granulated sugar, and brown sugar until slightly emulsified and glossy, about 1 minute. Whisk in the eggs one at a time, then mix in the vanilla and sour cream until smooth.
Combine: Add half of the dry mix to the wet ingredients and stir just to incorporate. Pour in all of the cocoa–coffee mixture and stir until mostly combined. Add the remaining dry mix and fold until only a few streaks of flour remain.
Fold in the sweet potato gently until evenly dispersed. Let the batter rest for 5 minutes to hydrate for a more even crumb.
Divide evenly between the prepared pans and smooth the tops. Bake at 340°F for 34–40 minutes, or until a toothpick inserted into the center comes out with a few moist crumbs attached.
Cool in pans for 15 minutes, then turn out onto a wire rack, peel away the parchment, and cool completely before frosting.
Make the frosting: Beat the softened butter on medium-high speed until pale and fluffy, 2 minutes. Add the cream cheese and beat 2 minutes more until very smooth. Mix in the cocoa (for the frosting), vanilla, and salt on low. Add powdered sugar 1 cup at a time, beating on low until incorporated, then on medium for 20–30 seconds to aerate. Thin with milk, 1 tablespoon at a time, until spreadable.
Assemble: If needed, level the cake layers. Place one layer on a board and spread about 1 cup frosting over the top. Set the second layer on, pressing gently to set. Apply a thin crumb coat around and on top; chill for 15–20 minutes to set. Finish with a generous final coat and smooth or create swoops with an offset spatula.