Adjust oven rack 6 inches from broiler element and heat broiler.
Heat 2 tablespoons vegetable oil in 12-inch broiler-safe skillet over medium heat until shimmering.
Add 1 poblano chile, stemmed, seeded, and chopped fine; three-quarters of 1 finely chopped onion; and ¼ teaspoon table salt and cook until softened, about 5 minutes.
Stir in 2 minced garlic cloves, 2 teaspoons ground cumin, and 1 teaspoon dried oregano and cook until fragrant, about 30 seconds.
Add 1 pound 93 percent lean ground chicken and cook, breaking up meat with wooden spoon, until no longer pink, 3 to 5 minutes.
Stir in 1 rinsed can black beans, 1 cup jarred salsa verde, ½ cup chopped cilantro, and 2 tablespoons water and bring to simmer.
Season with salt and pepper to taste.
Spread chicken and bean mixture into even layer and sprinkle with 8 ounces shredded Monterey Jack cheese.
Transfer skillet to oven and broil until cheese is melted and spotty brown, about 3 minutes.
Sprinkle with 1 stemmed and thinly sliced jalapeño chile, ½ cup cilantro leaves, and remaining finely chopped onion.
Serve with warmed corn or flour tortillas and lime wedges.
