Heat the oven to 180C/160C fan/gas 4. Butter and line three 20cm loose-bottomed sandwich cake tins with baking parchment.
To make the sponge, beat the butter and sugar together in a stand mixer or using an electric whisk for 5-8 mins or until light and fluffy. Whisk in the eggs, one at a time, beating well after each addition. Fold in the flour, baking powder and milk to loosen. Divide the mixture between three mixing bowls, weighing for accuracy if you like. Colour one of the batters blue and another pink, starting with a little colouring, then adding more for a deeper finish. Leave one of the bowls of batter uncoloured.
Get the three cake tins and bowls of batter ready in front of you. Spoon a heaped tablespoon of blue batter into the centre of the first tin, then pink in the second, and plain in the third. Working quickly so that it doesn’t spread too much, add another dollop of a different coloured batter on top of each one. Don’t wait for the batter to settle or spread, just keep alternating the colours until you run out. The weight of each layer will cause the batter to spread as you go, so very gently tap and shake the cake tins to even out the mixture, then bake in the centre of the oven for 25-30 mins or until firm to the touch. Turn out top-side down onto wire racks straightaway, then leave to cool completely.
For the icing, put the butter, icing sugar and vanilla seeds into a stand mixer fitted with the paddle attachment or put in a bowl and use an electric whisk. Beat for 5 mins or until fluffy and aerated. Add the cream cheese and beat briefly until just combined.
Place one of the cakes on a cake stand or serving plate and spread 2 tbsp of the buttercream on top, then repeat with the other sponges. Spread a smooth layer of the icing over the top and sides of the cake, using a large palette knife, leaving some slightly less iced spots for a ‘naked’ finish. Top with sprinkles and candles, if you like. Will keep in an airtight tin for three days.