Place the chocolate and milk in a thin-sided bowl and suspend over a gently simmering bain-marie, ensuring that the bowl does not touch the water. Melt the chocolate, stir until smooth, distribute in glasses.
Heat the milk to approx. 80 °C (do not boil), add the matcha and sugar, mix well, fill over the back of a spoon carefully into the glasses.
Heat the milk to approx. 50°C, transfer to a high-topped container, hold at a slight angle, froth with the mixer. While frothing, move the blender up and down to the surface of the milk. Divide the milk foam between the glasses. Grate chocolate over the glasses, decorate with matcha powder.