Heat a grill pan on the stove. Brush each peach half with melted butter, place face down on grill pan. You want a nice char which should take about 3-5 minutes. If you're grilling outside, you can also do this on a charcoal or gas grill. Lay the peach halves face down, directly on the grill. Keep watch because charcoal is much hotter, so the peaches may only take 2-4 minutes. The char adds a lovely flavor, and also softens the peaches a bit.
Set peaches aside to cool a bit.
When peaches are cooled enough to handle, cut them into slices.
Lay out 6 plates.
Arrange a small handful of arugula on each plate.
Lay a few peach slices on each plate, along with 2-3 slices of prosciutto.
In a small bowl, tear the balls of burrata into pieces. The cream will run out and that's ok!
Spoon a bit of the cheese (don't forget the creamy center, too!) over each plate.
Drizzle olive oil and balsamic vinegar over each plate. I usually just eyeball this, but it's about 2 tablespoons olive oil to 1 tablespoon balsamic vinegar. Taste and adjust depending on how much dressing you like.
Add a generous pinch of Maldon salt, a few twists of freshly ground pepper, and a few hazelnuts.
Serve immediately.