Heat large skillet over medium heat. Place chicken and salsa verde into a medium mixing bowl, stirring to combine. Melt butter in skillet and add one tortilla at a time. Layer one half of tortilla with cheddar cheese, mozzarella, chicken mixture, black beans, feta, and additional cheddar and mozzarella. Fold tortilla over filled layer and press with spatula. Cook both sides until golden brown. Remove from heat and cut quesadillas in half.