For the pastry cream, whisk the eggs with the water, then stir in the sugar mixed with the cornstarch until there are no lumps at all. Lumps at this stage never disappear later.
Heat the mixture slowly, stirring, until it thickens, then let it bubble for a few seconds so the cornstarch cooks out. Take it off the heat and set it aside to cool.
For the syrup, combine the water and sugar in a pot and bring to a boil. Set aside to cool to room temperature.
For the sponge, whisk the eggs, sugar and honey with a mixer for 7 minutes. It is ready when the mixture has tripled in volume and a drizzle from the beater sits on the surface for a couple of seconds before sinking.
Add the flour in two or three parts, folding gently with a spatula. Stop as soon as it disappears: every extra stroke knocks out the air you just whipped in.
Grease baking paper with butter and spread the batter over it in an even layer. Bake at 200 °C (392 °F) for 10 minutes, until the surface springs back when pressed lightly.
Remove from the oven and let it cool, but only until it is warm to the touch, not cold. A warm sponge rolls; a cold one cracks.
Brush the sponge with the syrup, then spread the pastry cream over it in a thin, even layer, leaving a clear border along one long edge.
Roll it up from the long edge, using the paper to help you keep it tight, and cut into 10 slices with a clean, sharp knife.