In a small bowl, mix the Dijon mustard and mayonnaise until smooth.
Spread the mixture evenly over one side of each tortilla.
Sprinkle half of the Swiss cheese over two tortillas.
Layer the chicken evenly over the cheese.
Top with slices of ham.
Sprinkle the mozzarella and remaining Swiss cheese over the ham.
Place the remaining tortillas on top and press gently.
Heat 1 tablespoon butter in a large skillet over medium heat.
Cook each quesadilla for 3 to 4 minutes per side, or until golden brown and the cheese has completely melted.
Transfer to a cutting board and let rest for 2 minutes.
Slice each quesadilla into wedges and garnish with chopped parsley if desired.
Serve warm with your favorite dipping sauce.