Preheat oven to 350°F. Lightly grease an ovenproof baking dish large enough to hold 10 rolled tortillas (about 9x13 inches).
Cook the chicken breasts until fully done (internal temperature 165°F) by your preferred method (poaching, baking, or use leftover/rotisserie chicken). Let cool slightly, then shred with two forks.
In a liquid measuring cup, whisk together honey, lime juice, chili powder, paprika, cumin, cayenne (if using), onion powder, garlic powder, salt, and black pepper. This mixture will be the marinade/sauce base (should be about 1 cup).
Measure 2/3 cup of the marinade and add it to the shredded chicken. Toss to coat, cover, and refrigerate for 30 minutes or up to several hours. Reserve the remaining marinade (about 1/3 cup).
In a medium bowl, combine the reserved marinade with the entire 16-ounce bottle of salsa verde. Stir to combine. If you prefer a smoother sauce, pulse with an immersion blender until smooth.
Drain and rinse the black beans. If using frozen corn, thaw and drain; if using fresh, ensure it is cooked. In a large bowl, combine the marinated shredded chicken, drained black beans, and cooked corn. Mix until evenly distributed.
Measure out 1 cup of the shredded Monterey Jack cheese and set it aside for topping. Add the remaining 2 cups of shredded Monterey Jack to the chicken mixture and stir to combine.
Lay out one flour tortilla at a time and divide the filling evenly among the 10 tortillas. Spoon the filling down the center of each tortilla, roll up tightly, and place seam-side down in the prepared baking dish.
Pour the salsa verde + reserved marinade mixture evenly over the rolled enchiladas. Sprinkle the reserved 1 cup of Monterey Jack cheese evenly over the top.
Bake at 350°F for 30 minutes, until sauce is bubbling and cheese is melted. For a browned top, place under the broiler 2–3 minutes—watch closely to avoid burning. Let rest 5 minutes, then serve topped with sour cream and fresh cilantro.