Place the parmesan and rosemary in the mixing bowl then grate 10 seconds/speed 10. Transfer to a bowl and set aside.
Place the water, milk, sugar and yeast in the mixing bowl then warm 3 minutes/37C/speed 2.
Add the flour, butter, 40g of the reserved parmesan and rosemary mixture, egg and salt then knead 5 minutes. Meanwhile, line two baking trays with baking paper and grease a large bowl.
Transfer the dough to the prepared large bowl, cover with cling film and leave to rise in a warm place for 30 minutes.
Divide the dough into 8 balls, place on the prepared baking trays and flatten slightly. Cover again and leave in a warm place for a further 30 minutes. Towards the end of the resting time, preheat the oven to 200C fan.
Brush the bread rolls with 1 tbsp of olive oil and sprinkle with the remaining 10g of the reserved parmesan and rosemary mixture. Bake the buns for 15 to 20 minutes until golden brown. Transfer to a wire rack and allow to cool.