Preheat oven to 200°F. Lay the tortillas on a clean work surface. Put 1/4 of the cheese, 1/4 of the chicken, and 3 slices of avocado on one side of each tortilla. Fold the tortillas in half and press gently to close.
Melt 1 tablespoon of the butter over medium heat in a 12-inch skillet or large griddle. Add two of the quesadillas and cook for 3 to 4 minutes, or until the cheese begins to melt and the tortilla starts to brown (use a spatula to peek underneath). Turn and cook the other side for 2 or 3 minutes or until golden brown. Put the quesadillas on a baking sheet and keep them in the warm oven while you cook the remaining quesadillas in the remaining 1 tablespoon butter.
Slice the quesadillas into quarters and serve with seasoned black beans, a salad, or salsa.