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. Reserve half of the mixture in a separate container for basting during grilling.
02 - Place the chicken breasts in a large resealable plastic bag or shallow dish. Pour the remaining half of the salsa verde mixture over the chicken, turning to coat evenly. Seal 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 375°F to 400°F). Lightly oil the grill grates using a paper towel dipped in vegetable oil to prevent sticking.
04 - Remove the chicken from the marinade, allowing excess to drip off, and discard the used marinade. Place the chicken breasts on the hot grill and cook, covered, for 5 to 7 minutes per side, until the internal temperature reaches 165°F and juices run clear when pierced.
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, about 1 to 2 minutes.
06 - Transfer the chicken to a serving platter using tongs. Sprinkle with chopped fresh cilantro and serve immediately with lime wedges alongside for squeezing.