Preheat the oven to 350°F (175°C) and grease a 9-inch round cake pan with melted butter, then dust with a little flour to prevent sticking.
In a large mixing bowl, whisk the eggs and sugar together with a hand whisk until the mixture becomes pale, thick, and triples in volume, about 5-7 minutes, producing a ribbons stage when lifted.
Gently fold in the vanilla extract and lemon zest using a rubber spatula, ensuring not to deflate the airy mixture.
Sift the all-purpose flour and baking powder together, then gradually fold it into the wet mixture in batches, mixing softly until just combined and smooth.
Pour the batter into the prepared cake pan and gently smooth the surface with a spatula. Tap the pan lightly on the counter to remove air bubbles.
Bake in the preheated oven for 20-25 minutes, or until the top is golden and a toothpick inserted into the center comes out clean. The cake will be slightly domed and spring back when lightly pressed.