Mix the yogurt with the cane sugar and vanilla until the cane sugar has completely dissolved. Grate the chocolate in the yoghurt (I always do this with the peeler) and stir in. If necessary, stir in some milk if the yogurt is too thick.
One more note on the amount of sugar:
Purchased stracciatella yoghurt usually contains 14-15% sugar. With 300 g of natural yoghurt, this corresponds to a sugar quantity of around 45 g. In my opinion this is way too cute. Even with 30 g (approx. 10% sugar content) the yogurt is still very sweet.
For us, the 30 g are just right, although the whole thing is too sweet for me again. Therefore, please adjust the amount of sugar to your own taste if necessary.