Place the milk for each hot chocolate in separate saucepans. Add the cocoa, dark chocolate and 2 tbsp of caster sugar to one saucepan. Add the white chocolate and 2 tbsp of caster sugar to the other saucepan. Bring both mixture to a simmer over a medium heat, whisking at the same time until smooth.
Add more caster sugar to both hot chocolates to adjust their sweetness if necessary. Divide the dark hot chocolate into two mugs and divide the white hot chocolate into two mugs. Garnish the white hot chocolate with a pinch of ground ginger if necessary.
Garnish with the star anise and a mixture of the buttons before serving.