Heat a grill or grill pan to medium-high and brush peach halves with olive oil.

Grill peaches cut-side down 3–4 minutes until char marks appear; cool and cut into wedges.

Thread peach wedge, basil, mozzarella, and tomato onto each skewer.

Whisk balsamic glaze with vinegar, salt, and pepper; drizzle over skewers just before serving.