- Whip the eggs with the sugar, citrus zest and vanilla with electric beaters or in a stand mixer fitted with the whip attachment on high speed until the eggs are thick and pale and hold a ribbon when the beaters are lifted, about 3 minutes.
- Sift the flour and baking powder together and add this to the eggs while beating on medium-low speed, mixing until blended. In a separate bowl, whisk the butter, milk and oil together. Spoon about a cup of the batter into the butter and whisk it in. Add this all back to the base batter and fold until blended. Cover and chill the batter for 2 hours, to set the butter.
- Preheat the oven to 375 °F (190 °C). Lightly grease and flour 2 madeleine pans, tapping out any excess flour.
- Stir the chilled madeleine batter gently to deflate it slightly (this is key to help getting the little bump that a well-made madeleine has) and with an ice cream scoop or a piping bag fitted with a plain tip, fill the madeleine molds (they will fill ¾ full). Bake the madeleines for 10 minutes, then bang the pans down (again, for the bump) before returning to the oven for another 5-7 minutes, until the madeleines are golden brown at the edges. Remove the madeleines from the pan immediately and enjoy warm or at room temperature.