In a medium bowl, whisk yogurt, chile, turmeric, garam masala, 3/4 tsp. chaat masala (if using), salt, lemon juice, oil, garlic, and ginger until well blended. Add paneer and mix with a spatula until evenly coated. Chill at least 2 hours.
Half an hour before grilling, soak 12 wooden skewers (about 10 in. long) in water. Heat a grill to high (450° to 500°). Onto each skewer, alternately thread a few pieces of onion (take chunks apart) followed by a chunk of peach, and paneer; repeat. Coat skewers well all over with cooking oil spray.
Oil grill grate, using tongs and a wad of oiled paper towels. Grill skewers until browned, turning once (if they stick, ease from grates with a metal spatula and tips of tongs), about 5 minutes total. Transfer to a rimmed baking sheet.
Thread a mint leaf onto ends of each skewer. Set on plates and serve with additional 2 tbsp. chaat masala for sprinkling, if using, and Cilantro Mint Chutney.
*Find Kashmiri chile and chaat masala at Indian markets and online. Find paneer, a low-fat Indian cheese, at well-stocked grocery stores and Indian markets.
MAKE AHEAD: Through step 1, up to 1 day.