In a large bowl, mix salsa verde, olive oil, lime juice, cumin, salt, and pepper to create the marinade.
Add the chicken breasts to the bowl and coat evenly. Cover and refrigerate for 30 minutes to 2 hours.
Preheat the grill to medium-high heat, about 400°F to 425°F.
Remove the chicken from the marinade and let excess marinade drip off.
Grill the chicken for 4 to 5 minutes per side until the internal temperature reaches 165°F.
During the final minute of grilling, place one slice of pepper Jack cheese on each piece of chicken and close the lid briefly to melt.
Transfer the chicken to a plate and let it rest for 3 to 5 minutes before serving.
Serve with fresh cilantro and lime wedges.