Preheat oven to 450°F (230°C) and line a 17×12-inch jelly roll pan with parchment paper.
Beat 2 eggs, 3 egg yolks, and ½ cup sugar on medium-high for 5 minutes until pale and thick.
Add vanilla extract and mix briefly.
Sift flour and cornstarch over the mixture and gently fold with a spatula.
Fold in the batter bits.
In a clean bowl, whip 2 egg whites and 1 tablespoon sugar until firm peaks form.
Gently fold the whipped egg whites into the batter.
Spread the batter evenly into the prepared pan.
Bake for 6–7 minutes until golden and springy to the touch.
Dust cake with powdered sugar, invert onto a towel, remove parchment, dust again, and roll while warm.
Cool the rolled cake on a wire rack.
For buttercream: whip butter 2–3 minutes; gradually add sugar; whip again 2 minutes.
Add vanilla and heavy cream until desired consistency, then fold in sprinkles.
Unroll cooled cake and spread buttercream evenly.
Roll the cake back up and refrigerate for a few hours.
Melt white chocolate and pour over the chilled cake.
Add sprinkles and allow chocolate to set before slicing.