Preheat oven to 350°F and line a jelly roll pan with parchment paper.
Beat egg yolks with sugar until thick and pale. Add espresso, vanilla, and melted butter.
Sift flour, baking powder, and salt. Fold gently into yolk mixture.
Whip egg whites until stiff peaks form, then fold into batter.
Spread batter evenly in pan and bake 12–15 minutes until springy.
Turn cake onto a powdered-sugar-dusted towel, peel off parchment, and roll with towel inside. Cool.
Beat mascarpone, powdered sugar, and vanilla until smooth. Add cream and whip to stiff peaks. Mix in espresso and liqueur.
Unroll cake, brush with espresso, and spread filling evenly. Re-roll without towel.
Wrap and chill for at least 2 hours or overnight.
Dust with cocoa powder and chocolate shavings before serving.