Preheat the oven to 350 degrees F. Grease a 9-by-13-inch baking dish with 1 tablespoon butter.
Put the ham on a paper towel-lined plate and microwave for about 3 minutes to reduce the moisture content. Mix together the Swiss and Monterey Jack cheese in a bowl.
Put about a third of chicken in a layer in the bottom of the prepared baking dish. Top with half the ham. Sprinkle over 1/4 cup of the mixed cheese. Repeat 1 more time, then finish with a layer of chicken.
Melt the remaining 5 tablespoons butter in a skillet over medium heat. Put the panko in a bowl, pour in half of the melted butter, mix well and set aside.
Add the flour to the remaining butter in the skillet and cook for 2 to 3 minutes, whisking out any lumps. Whisk in the milk, broth, Dijon and cayenne and let it cook until it begins to thicken. Remove from the heat and stir in 1 cup of the cheese until melted. Season with salt and pepper.
Pour the sauce over the casserole. Sprinkle over the remaining 1 cup cheese, then the panko mix. Bake until the top is browned and the casserole is bubbling, 25 to 30 minutes.