Heat a cast iron skillet over medium heat for 2 minutes. Add the oil and butter. As soon as the butter is melted, add the bread cubes. Toss to coat and cook, stirring occasionally, until the bread is slightly toasted, about 5 minutes. Add the ham and green onions and cook, stirring, for another minute.
In a large mixing bowl, whisk together the eggs, milk, cheese, Worcestershire sauce, salt, mustard, and pepper. Pour over the bread and press down gently with a spatula.
Bake for 20 minutes, or until golden brown and set. Let cool 5 minutes before serving.