1. Combine yogurt, curry, and 1 tablespoon lime juice in a large heavy-duty zip-top plastic bag; squeeze bag to mix. Cut 3 shallow slits in each chicken breast. Add chicken, cilantro sprigs, and garlic to bag, squeezing to coat chicken. Seal and marinate in the refrigerator for 2 hours, turning occasionally.
2. Combine remaining 2 tablespoons lime juice, nectarines, and next 6 ingredients (through red pepper) in a bowl; toss gently. Cover and set aside.
3. Preheat grill to medium-high heat.
4. Remove chicken from bag; discard marinade. Sprinkle chicken with salt; place chicken on grill rack coated with cooking spray. Cover and grill for 4 minutes on each side or until chicken is done. Let stand 5 minutes. Cut chicken across the grain into thin slices.
5. Place 1/3 cup nectarine chutney in center of each flatbread. Divide chicken evenly among flatbreads. Top each with 4 cucumber slices, 1/4 cup arugula, and about 2 1/2 tablespoons red onion; roll up. Cut each wrap in half diagonally.