In a saucepan over medium heat, melt 2 tablespoons of butter until bubbly. Whisk in 2 tablespoons of flour, cooking for about 1 minute until slightly golden. Gradually whisk in 1 cup of milk and add a chicken bouillon cube, continuing to whisk until the mixture thickens, about 5 minutes. Stir in 2 tablespoons of Dijon mustard, a dash of hot sauce if desired, and a generous ½ cup of Parmesan cheese until melted into a creamy sauce.

In a large skillet, melt enough butter to coat the bottom over medium heat. Lay one flour tortilla in the skillet, then top with a handful of shredded Swiss cheese for a melty base. Layer on slices of grilled chicken and deli ham, followed by a sprinkle of toasted breadcrumbs, and drizzle with the creamy Parmesan Dijon sauce. Finish by adding another layer of cheese and placing a second tortilla on top.

Increase the heat to medium and cook the quesadilla for about 3-4 minutes, or until the bottom is golden brown and crispy. Carefully flip it using a spatula to avoid spilling the insides. Cook the other side for an additional 3-4 minutes, ensuring the cheese melts thoroughly and the tortilla becomes a delightful, golden brown color.

Once both sides of the Chicken Cordon Bleu Quesadilla are perfectly crispy and the cheese is gooey, transfer the quesadilla to a cutting board. Allow it to cool for a minute before slicing it into wedges. Serve the quesadilla hot alongside the remaining creamy Parmesan Dijon sauce for dipping.