Preheat oven to 425°F.
Toss onion with one tablespoon oil and ¼ teaspoon each salt and pepper on a rimmed baking sheet. Roast until onion begins to soften, about 10 minutes.
Meanwhile, heat the remaining one tablespoon of oil in a large skillet over medium-high.
Pat chicken dry with paper towels and season with remaining ½ teaspoon of salt and pepper.
Add chicken to skillet and cook until golden brown, about 3 minutes.
Flip and continue to cook until golden brown, about 3 minutes more.
Transfer chicken to baking sheet with onion and roast until chicken is cooked through, 10 to 12 minutes.
Transfer chicken and onion to a platter.
Spoon salsa verde over top and top with cilantro and cheese.
Serve with tortillas and lime wedges, if desired.
