Preheat an outdoor grill at medium-high heat and lightly oil the grate.

Cut peaches in half and remove peach stones. Cut each half in half again.

Wrap prosciutto around each peach section; secure with small bamboo skewers.

Grill peaches, turning often, until prosciutto gets a little crisp around the edges and peaches begin to caramelize, about 6 minutes.

To serve, spoon burrata onto serving plates. Remove peaches from skewers; place several grilled peaches around cheese. Drizzle with olive oil. Top with a sprinkle of sea salt and pepper; garnish with basil.