Spray a 9'' x 13'' baking dish lightly with cooking spray. Then line the bottom of the dish with 1 layer of bread, cutting slices to fit if needed. Arrange half the prosciutto evenly over the bread, followed by half the goat cheese and half the provolone. Sprinkle with half of the green onions and basil.
Top with a second layer of bread slices. Layer on the remaining prosciutto, goat cheese, provolone, green onions, and basil.
Cut remaining bread slices into ½'' cubes to equal 2 cups of bread cubes. Sprinkle these over the top.
In a medium bowl, whisk eggs, milk, mustard, salt, and pepper until well combined. Slowly pour egg mixture evenly over the egg bake, letting it soak in. Drizzle melted butter evenly over the top. Cover and refrigerate overnight.
The next morning, preheat oven to 350°F. Uncover egg bake and let stand at room temperature for 30 minutes.
Bake until center of egg bake is set, the edges are bubbling a bit, and the top is browned, about 60 to 70 minutes. If you want to crisp up the top even more, place under broiler for a minute or two. Let rest for 5-10 minutes, then cut into large squares and serve.
