Preheat the oven to 425°F. Arrange a rack in the center of the oven.
Season the chicken evenly with pepper. Place the chicken in a 12-inch ovenproof skillet or 3-quart baking dish. Drizzle with the oil. Pour the salsa verde over the chicken.
Bake until the salsa is bubbly and a thermometer inserted into the thickest part of the chicken registers 140°F, 20 to 25 minutes.
Remove from the oven and turn the oven to broil. Sprinkle the chicken with the cheese and return to the oven. Broil until a thermometer inserted into the thickest part of the chicken registers 165°F, and the cheese is melted and browned in spots, about four minutes.
Season the chicken with salt to taste. Garnish with cilantro, if using.
