Preheat grill: heat grill or cast iron grill pan over high heat for at least 5 minutes.
Prep nectarines: halve and pit nectarines. Brush cut sides lightly with olive oil.
Grill: place cut-side down on hot grill. Cook 3 to 4 minutes without moving until grill marks form. Flip, cook 1 to 2 minutes skin side. Remove and rest 5 minutes.
Build the base: arrange arugula across a large serving plate. Scatter basil leaves on top.
Add nectarines: place grilled nectarines cut-side up on the greens.
Add burrata: tear burrata into large pieces and place between the nectarines. Let the creamy center spill out.
Drape prosciutto: fold prosciutto loosely over and around the cheese and fruit.
Finish and serve: drizzle olive oil, balsamic glaze, and honey over the plate. Add flaky salt, cracked pepper, and pine nuts. Serve immediately.