Preheat oven to 400 degrees F.
Spread 2 slices of bread evenly with mustard. Top each with 1/4 of the cheese. Place 1/2 of the ham and turkey on cheese. Cover with remaining cheese. Top with remaining slices of bread to form 2 sandwiches.
In a large shallow dish, whisk together eggs and milk; season with salt, pepper, and nutmeg. Dip each sandwich in egg mixture to coat well on both sides.
On a griddle or in a large saute pan, melt butter over medium heat. Add sandwiches and cook until golden brown, about 4 minutes per side. Cut each sandwich in 1/2 and serve immediately, dusted with confectioners' sugar, if desired.
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