Preheat oven to 350°F.
In a medium bowl, combine 1/4 cup cocoa powder, 1 tsp espresso powder, and 1/3 cup hot water. Whisk to combine.
Add sour cream, vegetable oil, and 2 egg yolks. Mix until cohesive and set aside.
In another bowl, sift together the 2/3 cup + 1 tbsp flour, the other 1/4 cup of cocoa powder, 1 tsp baking powder, 3/4 tsp salt, and 1/3 cup sugar.
In a stand mixer, whip your 2 egg whites with the remaining 1/3 cup of sugar until stiff peaks form.
Alternate adding your whipped egg whites and flour mixture to your chocolate and egg yolk mixture, ending with your egg whites.
Line a 16.25" x 12.25" jelly roll tray or baking sheet with parchment paper. Add your cake batter and cook until cake springs back, about 12-15 minutes.
While the cake bakes, dust a kitchen towel with a light coating of cocoa powder.
Cool cake for 1-2 minutes and place cake-side down onto the cocoa powder-dusted kitchen towel. Remove parchment paper. Roll your cake into a log with the kitchen towel. Let cool for 5-10 minutes in the kitchen towel, then, unravel carefully.
In a stand mixer, beat softened butter for 1-2 minutes. Add your powdered sugar and mix on low to combine.
Add your peppermint extract, milk, and 1/2 tsp salt.
Top your chocolate cake with your peppermint buttercream, leaving a 1-2 inch border around the top and bottom.
In a large bowl or plastic bag, smash your candy canes with a mallet into large chunks.
Sprinkle the top of your frosting layer with candy cane shards. Save the rest of your shards for later in a covered bowl. Then, roll your cake back into a log, removing the kitchen towel as you roll (like making sushi).
Slice about 1/2" to 1" of cake off of each end of your log and set aside scraps. Then, slice your roll into 1" slices.
Set slices on a plastic-wrap-lined baking sheet and place in the freezer, covered, for 5-10 minutes until firm. Then, transfer to fridge.
In a microwave-safe bowl or double boiler, heat chocolate gently until fully melted. You may need to add 1-2 tbsp of butter to help it achieve a runny consistency. Remove from heat.
One at a time, dip your chilled Swiss roll slices in chocolate, only covering the left edge. Then, dip into your reserved candy cane shards and press to stick.
Place Swiss rolls back in the fridge to set the chocolate. Serve at room temperature.