6 tablespoons butter, plus more for greasing dish
1 large onion, thinly sliced
Kosher salt and freshly ground black pepper
4 cups 1/2-inch bread cubes (crusts removed)
8 ounces cremini mushrooms, sliced
8 large eggs
1 1/4 cups half-and-half
1 teaspoon Dijon mustard
Pinch cayenne
4 ounces ham, diced
1 1/2 cups shredded Gruyere, Emmental or Jarlsberg (about 4 ounces)
2 small plum tomatoes, sliced very thinly
2 tablespoons grated Parmesan
2 tablespoons chopped parsley, for garnish