Preheat a grill pan or outdoor grill to medium-high heat.
Lightly brush peach slices with olive oil and grill for 2 minutes per side until char marks appear and fruit slightly softens.
In a large mixing bowl, toss baby arugula with remaining olive oil, sea salt, and black pepper until evenly coated.
Divide the seasoned arugula between two serving plates as a base.
Arrange the grilled peach slices, sliced chicken breast, and mozzarella pearls on top of the greens.
Tear the prosciutto slices into thin ribbons and distribute them evenly throughout the salad.
Garnish both servings with toasted pistachios for texture.
Drizzle the balsamic glaze in a cross-hatch pattern over the assembled salads and serve immediately.