Preheat toaster oven to 425°. Thinly slice green onions; reserve 2 tablespoons dark green tops for garnish.
Evenly divide chicken, cheese, jalapeño, cilantro, and remaining green onions on one side of each tortilla; fold other half over.
In single layer on foil-lined toaster oven tray (working in batches if necessary), toast quesadillas 8-10 minutes, or until tortillas are golden brown and cheese is melted.
Cut each quesadilla in half. Serve with salsa verde, add a dollop of sour cream if you’d like, and sprinkle with reserved green onions.