Heat oven to 375°F. Line a 10×15-inch jelly-roll pan with parchment and lightly coat it.
Whisk flour, cocoa, baking powder, salt, and espresso powder in a bowl.
Beat eggs and granulated sugar on high until thick and pale, about 3–4 minutes.
Mix in vanilla, oil, and milk just until combined.
Sift the dry mixture over the batter and gently fold until no dry pockets remain.
Spread batter evenly in the pan and smooth the top. Tap the pan to release bubbles.
Bake until the cake springs back lightly at the center, 10–12 minutes.
Dust a clean kitchen towel with powdered sugar. Invert the hot cake onto the towel and peel away parchment.
Roll the cake up with the towel from a short end. Let it cool for 20 minutes.
Beat mascarpone, powdered sugar, and vanilla until smooth. Stream in cold cream and whip to medium peaks.
Unroll the cake. Spread the cream evenly, leaving a 1/2-inch border. Re-roll and chill seam-side down for 20 minutes.
Warm cream until steaming. Pour over chopped chocolate and butter, rest 2 minutes, then stir glossy. Cool until spreadable.
Frost the chilled roll with ganache. Drag a fork for bark texture, dust with powdered sugar, and garnish with raspberries.