In a large bowl or gallon-sized ziplock bag, combine salsa verde, olive oil, lime juice, cumin, salt, and black pepper. Whisk or shake until well blended and the oil is fully incorporated.

Add the chicken breasts to the bowl or bag and toss or massage thoroughly to ensure every piece is completely coated with marinade. If using a bag, squeeze out excess air and seal. Cover the bowl with plastic wrap if using that method. Refrigerate for at least 30 minutes, or up to 2 hours. Flip the chicken halfway through marinating time for even flavor distribution.

About 10 minutes before cooking, preheat the grill to 400-425°F (medium-high heat). Clean the grates and lightly oil them with a paper towel dipped in vegetable oil.

Remove the chicken from the marinade, let excess drip off, and discard remaining marinade. Pat chicken lightly with paper towels to remove excess moisture for better grill marks.

Place chicken on the preheated grill and cook for 4-5 minutes without moving. Flip when edges turn white and chicken releases easily from grates. Cook another 4-5 minutes until internal temperature reaches 165°F. Avoid pressing down on the chicken.

During the final minute of cooking, place one slice of pepper Jack cheese on each chicken breast and immediately close the grill lid. Cook for 45-60 seconds until cheese is fully melted and just starting to bubble.

Remove chicken from grill and transfer to a clean plate. Let rest for 3-4 minutes before serving to allow juices to redistribute. Garnish with fresh cilantro and serve with lime wedges if desired.