In a large mixing bowl, whisk together the salsa verde, olive oil, lime juice, minced garlic, ground cumin, dried oregano, salt, and black pepper to create the marinade.

Add the chicken breasts to the bowl, ensuring they are fully coated in the marinade. Cover the bowl and refrigerate for at least 30 minutes, or up to 4 hours for maximum flavor. (Do not marinate for more than 4 hours, as the lime juice can start to break down the chicken's texture).

Preheat your grill to medium-high heat (around 400-450°F or 200-230°C). Lightly oil the grill grates to prevent sticking.

Remove chicken from the marinade, allowing any excess to drip off. Discard the remaining marinade.

Place the chicken breasts on the preheated grill. Grill for 6-8 minutes per side, or until the internal temperature reaches 165°F (74°C) with a meat thermometer.

During the last 2-3 minutes of grilling, place one slice of pepper jack cheese on top of each chicken breast. Close the grill lid to allow the cheese to melt and become bubbly.

Once the chicken is cooked through and the cheese is melted, carefully remove the chicken from the grill and transfer it to a clean cutting board or serving platter.

Tent the chicken loosely with foil and let it rest for 5 minutes before serving. This helps to redistribute the juices, ensuring a more tender and flavorful chicken.

Garnish with fresh chopped cilantro, if desired. Serve hot with your favorite sides like rice, black beans, or a fresh salad.