To make the sandwiches: Lay the bread out on the work surface. Spread one side of each slice with the mustard. Spread half of the cheese on 4 slices of the bread. Cover with half of the turkey and season with salt and pepper to taste. Evenly spread the cranberry on the turkey on the sandwich. Top with the remaining turkey; season with salt and pepper to taste, and cover with the remaining cheese and bread. Press down slightly on each sandwich.
In a lipped plate or dish, whisk together the eggs and milk, and season with salt and pepper to taste.
Preheat the oven to 350 degrees F. Heat 2 large skillets over medium-low heat, add a tablespoon of both butter and oil to each pan. Dip and coat each of the sandwiches in the egg mixture and lay in the pans. Cook until golden brown, about 4 minutes per side. Transfer the sandwiches to a baking sheet and bake until the cheese is melted, about 6 to 8 minutes.