Whip powdered sugar, butter, cocoa powder, salt, and coffee liqueur together in the bowl of a stand mixer on high speed. Transfer buttercream into a separate bowl and add mascarpone cheese. Mix until combined; set aside.
Preheat the oven to 400 degrees F (200 degrees C). Butter a 13x18-inch rimmed sheet pan. Line pan with parchment paper and brush melted butter on top.
Combine cocoa powder, salt, and flour together in a bowl; whisk or sift to break up clumps.
Place eggs in the clean bowl of your stand mixer. Add sugar and whip until fluffy, thick, and very light in color. Add 1/2 of the cocoa powder mixture and vanilla extract; mix on low speed for a few seconds. Beat in remaining cocoa mixture on low for a few seconds. Switch to high speed; stop once mixture is moistened but not fully blended. Pull off the whisk attachment and whisk batter with it until evenly blended.
Pour batter onto the prepared sheet pan and spread out with a spatula, leaving some room around the edges. Tap pan on the counter to knock out the large bubbles.
Bake in the preheated oven until top is dry and edges start to pull away from the sides, 8 to 10 minutes.
Dust a clean kitchen towel with enough powdered sugar to cover an area slightly larger than the sponge cake. Remove cake from the oven. Run a knife around the edges of the pan. Sprinkle some powdered sugar over the top. Run a spatula under the parchment paper to make sure it's not stuck to the pan.
Quickly flip pan on top of the sugared area to invert the cake. Remove parchment paper and dust cake with more powdered sugar. Gently roll cake up inside the towel; allow to cool for 15 minutes.
Unroll cake and dollop buttercream on top, reserving some for later. Spread frosting to the edges. Roll cake up over the frosting, using the towel to lift it if needed. Sprinkle more powdered sugar on top. Wrap log in plastic wrap. Refrigerate until firm, about 2 hours.
Combine chocolate chips and hot cream in a bowl. Let sit for 1 minute. Whisk until chocolate melts.
Make an angled cut 3 inches from one end of the log. Place log on a parchment-lined sheet pan. Apply some buttercream to the angled slice and attach it to one side. Spread a layer of ganache all over the cake, except for the swirls. Refrigerate for 15 minutes to firm up ganache.
Carve lines into the ganache using the tip of a knife to create the appearance of tree bark. Refrigerate until completely chilled before serving. Dust with cocoa powder and powdered sugar.