Preheat the oven to 160°C (fan assisted).
Lightly grease two 7inch round cake tins with butter or spray oil. Line the base and sides with greaseproof paper.
Break up both chocolates into separate bowls and melt in the microwave in 30-second blasts, stirring in-between, until smooth. Set aside.
In a small bowl, mix the cocoa powder, espresso powder and boiling water together to make a chocolatey paste. Set aside.
In a large bowl, cream the butter, sugar and vanilla together until light and fluffy. Beat in the eggs one at a time until well combined. Now fold in the flour and milk to make a smooth cake mixture.
Divide the mixture evenly between two bowls. Fold the melted white chocolate into one bowl. Fold the melted milk chocolate and chocolatey paste into the other bowl.
Spoon the cake mixtures into the baking tins, smoothing out the tops. Bake for 35-40 minutes until risen and springy to the touch. Leave to cool completely.