Grease a springform pan (20 cm Ø) for the dough. Preheat the oven (electric cooker: 175 °C/circulating air: 150 °C/gas: see manufacturer). Wash lemon hot and dab dry. Finely grate the peel. Squeeze lemon.
Cream 125 g soft butter, cream cheese, lemon zest, vanilla pulp, sugar and 1 pinch of salt with the whisks of the mixer. Stir in the eggs one by one. Mix flour and baking powder. Stir 2 tablespoons lemon juice into the sugar-egg mixture.
Smooth the dough in the form. Bake in a hot oven for 45-50 minutes. Remove from the oven. Let it cool down.
For the icing, sieve icing sugar into a bowl and mix with milk to a smooth icing. Remove the cake from the mould. Spread the icing on the cake and let it dry.