Preheat & prep
Heat the grill to medium-high. Oil a paper towel with grapeseed or olive oil and wipe the grates so the peaches don’t stick. Slice the peaches into 1/2-inch wedges. Quarter the vegan mozzarella and get the pesto ready.
Grill the peaches
Toss peach slices lightly with a bit of olive oil and a pinch of salt. Place them on the hot grill. Nudge them after about 1 minute so they don’t stick. Flip after about 3 minutes, once grill marks form and caramelization appears. Grill another 2–3 minutes until slightly soft. Move to a plate and cover to keep warm.
Grill the flatbread
Brush each naan lightly with oil or use a quick spray. Grill each naan 2 minutes per side until warm and you see grill marks. Remove from the grill and spread ½ cup arugula pesto across both flatbreads (about ¼ cup each).
Assemble
Top each pesto-covered flatbread with grilled peach slices, half the vegan mozzarella, and a handful of fresh arugula. Drizzle with balsamic glaze if you like a sweet-tangy finish. Slice and serve immediately.