Preheat the oven to 375 degrees F (190 degrees C). Lightly grease a 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 spread out in an even layer on a baking sheet. Bake until lightly browned and toasted for 12-14 minutes.

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 and tear half of the ham slices into 1 ½-inch pieces and disperse between bread cubes. Top evenly with 2 cups grated Swiss cheese.

Top with remaining bread, ham, and grated Swiss cheese. Pour egg mixture over casserole and press lightly to submerge bread. Let stand at room temperature for 30 minutes.

Bake casserole until egg mixture is absorbed, for 45-50 minutes. Cover with foil during last 20 minutes if browning too quickly.

While casserole bakes, make cream sauce. Melt remaining butter in a saucepan over medium heat, whisk in flour, and cook for 1 minute. Gradually whisk in milk until smooth. Simmer until thickened and cooked, about 8 minutes. Add remaining salt, pepper, and nutmeg, then remove from heat.

Remove casserole from the oven, turn on the broiler, and pour cream sauce on top. Sprinkle Parmesan and broil until bubbly and lightly browned for about 3 minutes.