Line bottom of 13x9x2-inch glass baking dish completely with 1 layer of bread, cutting some slices to fit. Arrange half of prosciutto evenly over bread. Sprinkle half of goat cheese and half of provolone over. Sprinkle with half of green onions and half of basil. Top with second layer of bread. Layer remaining prosciutto, goat cheese, provolone, green onions, and basil atop bread. Cut remaining bread into 1/4-inch cubes. Sprinkle over top.
Whisk eggs, milk, mustard, and salt in bowl. Season with pepper. Pour egg mixture over strata; press down on bread with spatula. Drizzle melted butter over strata. Cover and refrigerate overnight.
Preheat oven to 350°F. Uncover strata and let stand at room temperature 30 minutes. Bake until center is set, about 1 hour. Remove from oven.
Preheat broiler. Place strata under broiler until top is golden, about 30 seconds.
Cut into large squares and serve.