Mix the shredded chicken, cream cheese, hot sauce, blue cheese, Monterey jack, and green onions in a medium-sized bowl. Stir until the mixture is well combined.
Separate the wontons, then place one wrapper in front of you on the counter or a cutting board.
Place approximately one tablespoon of the filling in the middle of the wrapper, then dip your finger in the water dish and moisten the wrapper's edges.
Fold one half of the wrapper over the filling to form a triangle, then gently press the edges together to seal them.
Fold the two points of the long edge of the triangle over the filling to meet in the middle and press them together. You will have a little pillow, sort of resembling an envelope.
Place the prepared wontons in a single layer in your air fryer basket, and then give them a generous spray of olive oil.
Set the air fryer to 380° and cook the wontons for 7-10 minutes until crispy and golden, turning halfway through.
Serve garnished with additional crumbled blue cheese or sliced green onion with dishes of blue cheese dip, hot sauce, or ranch, if desired.