Grease a 9x13-inch baking dish. Spread the bread cubes in the dish and top with the ham and 1 cup of the cheese.
In a large bowl, whisk the eggs, milk, mustard, salt and pepper until blended.
Pour the egg mixture evenly over the bread. Press down gently so the bread soaks it up, then sprinkle with the remaining ½ cup cheese.
If making ahead, cover and refrigerate overnight. Otherwise let it stand for 15 minutes while the oven heats.
Preheat the oven to 350°F. Bake uncovered for 40–45 minutes, until puffed, golden and set in the center.
Let it rest for 5 minutes, top with the green onions and serve.