Preheat your grill to 375°F and oil the grates well to prevent sticking. While the grill heats, slice the peaches into 1/2-inch thick wedges, toss the arugula with the fresh lemon juice and a pinch of salt in a small bowl, and have the burrata torn into 1-inch pieces ready on a plate. This prep work ensures everything moves smoothly once grilling begins.
Lightly brush the peach wedges with 1 tablespoon of olive oil on both sides. Place them on the hot grill and cook for 3-4 minutes per side until you see beautiful caramelized grill marks. The peaches will soften slightly and develop a sweet, smoky flavor—I like to grill them first so they're ready to add at the end without any cooling down.
Brush both naan flatbreads with 1 tablespoon of olive oil on one side. Place them oil-side down on the grill and toast for 2-3 minutes until light golden and marked. This initial grilling creates a crispy base and adds a subtle charred flavor that contrasts beautifully with the creamy toppings.
Flip the naan over and immediately spread 3 tablespoons of pesto evenly across each flatbread. Distribute the torn burrata pieces from Step 1 generously over the pesto layer. Lower the grill heat slightly if needed, cover the grill with the lid, and cook for 3-4 minutes until the burrata begins to melt and soften—it should be creamy but not completely dissolved.
Transfer the flatbreads to a serving platter or individual plates. Arrange the grilled peach wedges from Step 2 on top of each flatbread, scattering them evenly. Top with the dressed arugula from Step 1, then sprinkle with the cracked black pepper and flaky sea salt. Drizzle with honey and balsamic glaze (if using) for a sweet-savory finish—I find the combination of warm, melted cheese with cool arugula and sweet peaches creates the perfect balance of temperatures and flavors.