01 - In a medium bowl, whisk together the salsa verde, olive oil, ground cumin, garlic powder, smoked paprika, kosher salt, and black pepper until well combined. Set aside half of the mixture for basting during grilling.
02 - Place the chicken breasts in a large resealable plastic bag or shallow dish. Pour the remaining salsa verde mixture over the chicken, seal or cover, and refrigerate for at least 30 minutes or up to 4 hours for deeper flavor penetration.
03 - Preheat an outdoor grill or grill pan to medium-high heat, approximately 400°F. Lightly oil the grates with vegetable oil to prevent sticking.
04 - Remove the chicken from the marinade and discard the used marinade. Place the chicken on the hot grill, cover, and cook for 5 to 7 minutes per side until the internal temperature reaches 165°F and juices run clear.
05 - During the final 2 minutes of grilling, spoon the reserved salsa verde mixture generously over each chicken breast. Top evenly with shredded Pepper Jack cheese, close the grill lid, and allow the cheese to melt completely.
06 - Transfer the chicken to a serving platter. Garnish with chopped fresh cilantro and serve immediately with lime wedges on the side.