Special equipment: a metal fish spatula
For the Dijon dressing: Lightly whisk the extra-virgin olive oil, vinegar and mustard together in a small bowl. Season with salt and pepper and set aside.
For the gilled peaches: Lightly grease a griddle pan or barbecue with the olive oil and then get it nice and hot. Place the peach quarters cut-side down on the griddle pan or barbecue for 1 to 2 minutes to get a nice chargrilled mark on them. You may have to work in batches, depending on the size of your pan.
Use a metal fish spatula to help lift them up so as not to leave the chargrilled marks behind. Turn them onto their other cut side for another minute (don't cook on their skin side). The charring will add to the flavour. You can also use a frying pan to cook them, but obviously you won¿t get the chargrilled lines on them. Sit the peaches cut-side up on a large serving platter as you go.
Arrange the buratta chunks (or mozzarella) around the peaches and scatter the arugula and basil leaves over. Scatter a little salt and pepper on top. Quickly whisk up the dressing again, if necessary, drizzle it over and serve.