Prepare the pan: Preheat oven to 375°F (190°C). Line a 15x10-inch jelly roll pan with parchment paper, leaving 2 inches overhang on each side. Lightly grease the parchment paper.
Make the chocolate sponge: In a large bowl, beat eggs and granulated sugar with electric mixer on high speed for 5 minutes until thick, pale, and tripled in volume. The mixture should fall in ribbons when beaters are lifted.
Add dry ingredients: Sift together flour, cocoa powder, and salt. Gently fold dry ingredients and melted butter into egg mixture using a rubber spatula until just combined. Don't overmix.
Bake the cake: Spread batter evenly in prepared pan. Bake for exactly 12 minutes until cake springs back lightly when touched and pulls slightly from edges.
Roll immediately: Turn hot cake onto a clean kitchen towel dusted generously with powdered sugar. Peel off parchment paper and roll cake in towel starting from short end. Cool completely rolled up, about 1 hour.
Prepare filling: Beat heavy cream, powdered sugar, and vanilla with electric mixer until soft peaks form. Don't overwhip.
Assemble and chill: Carefully unroll cooled cake and spread filling evenly, leaving ½-inch border. Re-roll cake without towel. Wrap tightly in plastic wrap and refrigerate 1 hour before slicing.
