In a food processor, pulse the chicken, cheese, and broccoli until they just combine, but not so long that they turn into a gooey mess.
Move to a bowl, and stir in the cilantro, green onions, and tomatoes.
Place the filling inside the tortillas, whichever size you are using. If you are making mini-quesadillas, saute to order. For regular-size, grill or saute until the tortilla is crispy brown and the cheese is melted.
Serve with a dollop of spicy aioli (optional).