In a large bowl, whisk together the salsa verde, olive oil, lime juice, cumin, salt, and black pepper until well combined.

Add the thinly sliced chicken breasts to the bowl, tossing them until fully coated in the marinade. Cover with plastic wrap and refrigerate for 30 minutes up to 2 hours.

Preheat your grill to medium-high heat, aiming for about 400°F, and lightly oil the grill grates.

Remove the chicken from the marinade and place on the grill, cooking for 4-5 minutes on each side until the internal temperature reaches 165°F.

In the final minute of grilling, top each piece of chicken with a slice of pepper Jack cheese and close the grill lid.

Remove the chicken from the grill and let it rest for a few minutes before serving.