Grill the peaches: Preheat a grill or grill pan over medium-high heat. Brush the cut sides of the peaches with the olive oil and grill for 2–3 minutes per side, until grill marks appear and the peaches are slightly softened. Let cool for a few minutes, then slice into wedges.
Dress the arugula: In a medium bowl, toss the arugula with the olive oil, lemon juice, a pinch of kosher salt, and a few cracks of black pepper until lightly coated.
Toast the ciabatta: Lightly toast the ciabatta rolls until the outside is crisp and the inside is warm.
Assemble the sandwiches: Tear the burrata and spread it generously over the bottom halves of the toasted ciabatta rolls. Layer with the grilled peach slices, followed by the prosciutto and the dressed arugula. Drizzle with the balsamic glaze and chili crisp, then finish with a sprinkle of flaky sea salt and freshly cracked black pepper. Top with the remaining ciabatta halves.
Serve immediately while the peaches are still slightly warm and the ciabatta is crisp.