Preheat the oven to 375 degrees F (190 degrees C). Lightly grease a 9x13-inch baking dish. Melt 6 tablespoons butter.
Place English muffin cubes in a large bowl and top with melted butter and Dijon mustard. Toss until well coated and place spread out in an even layer on a baking sheet. Bake in the preheated oven until lightly browned and lightly toasted, 12 to 14 minutes.
Meanwhile, whisk eggs, half-and-half, 1 teaspoon salt, and 1/2 teaspoon pepper together in a bowl until well combined; set aside.
Place half of bread cubes in the prepared baking dish. Fold half of the ham slices and tear into 1 ½-inch pieces and disperse ham between bread cubes. Top evenly with 2 cups of grated Swiss cheese.
Top with remaining bread, ham, and grated Swiss cheese. Pour egg mixture over casserole and press lightly to submerge most of the bread. Let stand at room temperature for 30 minutes.
Bake casserole until egg mixture is absorbed, 45 to 50 minutes. Cover with foil during the last 20 minutes of baking time if casserole is browning too quickly.
Meanwhile, melt remaining butter in a saucepan over medium heat. Whisk in flour and cook, stirring constantly, about 1 minute. Whisk in milk until smooth. Bring mixture to a boil and reduce heat to simmer, stirring occasionally until thickened, about 3 minutes. Add in remaining salt, pepper, and nutmeg and stir. Cook about 8 minutes on low, stirring occasionally, and remove from heat.
Remove casserole from oven and turn on the oven’s broiler. Pour cream sauce over casserole and top with grated Parmesan. Place casserole back in oven and broil until bubbly and lightly browned, about 3 minutes. Let stand for 5 to 10 minutes. Sprinkle with chives and serve.
