Grilled Salsa Verde Chicken

In a large bowl, whisk together ½ cup salsa verde, olive oil, lime juice, garlic, chili powder and cumin. (Reserve about ½ cup salsa verde for basting chicken and ½ cup for serving.) Add chicken and toss until fully coated in mixture. Let marinate for 20 minutes. Preheat grill to medium-high, then add chicken and cook for 6 minutes. Flip chicken, brushing each breast with more salsa verde, and cook for another 6 minutes. Top each piece of chicken with a slice of Monterey Jack, jalapeño and red onion, then cover grill and cook until the cheese has melted and the chicken is cooked through, about 4 minutes more.