Melt the butter over a low heat. Once at the boil, leave for 5-10 minutes until it turns golden brown, with the brown residue visible on the bottom of the pan. Turn off the heat and strain the butter through a thin sieve. Leave to cool.
Mix the egg yolks with the sugar. They should triple in volume, at least. Add the flour and the melted butter and mix gently.
Heat up the milk and vanilla extract and gradually add it to the mixture, mixing continuously. The consistency will be fairly thin.
Whisk the egg whites with a pinch of salt until they become stiff. Add to the mixture with a spatula mixing gently. Do not overmix, even if the mixture is not fully smooth.
Preheat the oven to 160 °C (320 °F). Butter a 24/24cm (9.5/9.5 inch) non-stick baking tray. Pour the mixture into the tray. Bake for 45-50 minutes, until a nice golden brown.
Leave the cake to cool in the tray for 1-2 hours and then slice it into delicious cubes. Sprinkle with powdered sugar and serve.