Heat oven to 325ºF. Prepare a large baking dish(9x13") by coating with butter.
With each slice of bread, coat one side in softened butter, and the other side with a thin layer of mustard. Place butter side up in prepared pan. Arrange shingled over the bottom of the pan. Scatter chopped ham over top.
In a medium-sized bowl, whisk together milk, half-and-half, eggs, black pepper, and salt. Pour evenly over the bread mixture. Sprinkle the shredded cheese over top. Cover with plastic wrap and refrigerate for at least one hour or overnight, until the mixture is absorbed into the bread.
Preheat oven to 325ºF. Bake until the custard has set and the bread and cheese are golden brown. Serve warm and enjoy!