Stir together chicken, ginger-garlic paste, lime juice, yogurt, tandoori masala, oil, vinegar, chile powder, salt, and garam masala in a large bowl until evenly coated. Cover and refrigerate for at least 1 hour or up to 4 hours.
Preheat grill to high (450°F to 500°F). Thread chicken evenly onto skewers, leaving about 1 inch of space at top and bottom of each skewer; discard marinade. Brush grill grates with oil. Place skewers on oiled grates over direct heat; grill, covered, until grill marks form and chicken releases easily from grates, 3 to 4 minutes. Turn skewers; grill, covered, until lightly charred and a thermometer inserted into thickest portion of chicken registers 165°F, 5 to 6 minutes. Transfer skewers to unlit side of grill to keep warm.
Brush 1 side of each flatbread evenly with ghee. Place flatbreads, buttered sides up, on oiled grates over direct heat. Grill, uncovered, until lightly charred on bottom, 30 seconds to 1 minute. Flip bread, and grill until lightly charred and warmed through, about 1 minute.
Slide chicken evenly off skewers onto buttered sides of each flatbread. Top evenly with red onion and cilantro; drizzle evenly with green chutney. Roll tightly, and serve immediately.
