Preheat griddle on appropriate setting for 5 minutes.
Spread four of the bread slices with mustard, layer smoked turkey and cheese on top; place remaining four bread slices over each to make sandwiches.
In a 9-inch pie plate, whisk together eggs, milk, salt, and pepper.
Dip sandwiches in egg mixture, turning to coat both sides.
Lightly grease griddle with butter.
Place sandwiches, cheese-side down, on griddle; cook 4 to 9 minutes or until golden brown, turning every 2 minutes.