Preheat the oven to 350°F (175°C). Line a 10-by-15-inch jelly roll pan with parchment paper, leaving a little overhang on the short sides, and lightly grease the parchment.

Whisk the flour, cocoa powder, baking powder, and salt together in a medium bowl. Sifting these ingredients helps remove lumps and keeps the delicate sponge light.

Place the eggs and granulated sugar in a large mixing bowl. Beat with an electric mixer on medium-high speed for 4 to 5 minutes, until the mixture is very pale, thick, and fluffy, then beat in the vanilla.

Sift the dry ingredients over the egg mixture in two additions. Fold gently with a spatula after each addition, using broad strokes from the bottom of the bowl so the whipped eggs do not lose too much air.

Drizzle the cooled melted butter around the edge of the bowl and fold just until incorporated. Spread the batter evenly in the prepared pan, making sure it reaches the corners.

Bake for 10 to 12 minutes, until the top springs back lightly when touched and a toothpick inserted in the center comes out with a few moist crumbs. Do not overbake, because a dry sponge is more likely to crack.

While the cake bakes, place a clean kitchen towel on the counter and dust it generously with powdered sugar. As soon as the cake comes out of the oven, run a thin knife around the edges, invert the cake onto the towel, and carefully peel away the parchment.

Starting at one short end, roll the warm cake and towel together into a snug spiral. Set it seam-side down on a wire rack and let it cool completely, about 30 minutes.

For the filling, beat the cold heavy cream, powdered sugar, and vanilla in a chilled bowl until medium-stiff peaks form. The cream should hold its shape but still look smooth and spreadable.

Carefully unroll the cooled cake and spread the whipped cream evenly over the surface, leaving a 1/2-inch border around the edges. Roll the cake back up without the towel, keeping the spiral firm but not tight enough to squeeze out the filling.

Place the cake seam-side down on a serving platter. Chill for at least 30 minutes, then dust with powdered sugar, scatter chocolate curls over the top, slice with a sharp serrated knife, and serve.