Preheat oven to 400°F and lightly oil a 9x13-inch baking dish.
Stir cream cheese, sour cream, green chiles, garlic, cumin, chili powder, onion powder, lime juice, and a pinch of salt in a bowl until smooth.
Fold in 1 cup Monterey Jack and half the cilantro; reserve remaining cheeses for topping.
Pat chicken dry. Season both sides with salt and pepper.
Heat olive oil in a large skillet over medium-high. Sear chicken 2–3 minutes per side until lightly browned.
Arrange chicken in the baking dish. Spread the green chile mixture evenly over each piece.
Top with remaining Monterey Jack and the cheddar.
Bake 16–20 minutes, until chicken is cooked through and reaches 165°F internally.
Broil 1–2 minutes to brown the cheese, if desired. Rest 5 minutes, then garnish with cilantro and green onion.