Prepare the pan – Grease a 10x15-inch jelly roll pan, line it with parchment paper, and lightly grease again.
Mix the eggs and sugar – Beat the eggs with an electric mixer until pale and fluffy. Add sugar and vanilla. Mix until thick.
Add dry ingredients – In another bowl, whisk flour, baking powder, and salt. Gently fold this into the egg mixture, alternating with milk.
Bake the sponge – Pour the batter into the prepared pan. Spread evenly. Bake at 375°F (190°C) for 10–12 minutes until it springs back when touched.
Roll while warm – Dust a clean towel with powdered sugar. Flip the cake onto the towel, peel off the parchment, and roll it up gently from the short side. Let it cool rolled.
Add filling – Once cooled, unroll the sponge, spread jam evenly across the surface, then roll it back up (without the towel).
Serve – Place seam-side down, dust with powdered sugar, and slice into spirals.