Arrange bread pieces in the prepared baking dish; drizzle with melted butter. Top with Swiss cheese, Monterey Jack cheese, and ham.
Whisk milk, eggs, green onions, sherry, mustard, black pepper, and cayenne pepper together in a bowl until foamy; pour over bread mixture. Cover dish tightly with aluminum foil and refrigerate for 24 hours.
Remove dish from refrigerator 30 minutes before baking.
Preheat oven to 325 degrees F (165 degrees C).
Bake strata in the preheated oven for 1 hour. Remove aluminum foil and top with sour cream and Parmesan cheese. Bake, uncovered, until a knife inserted in the strata comes out clean, about 10 more minutes.