Toss together the Brussels sprouts, ham, mushrooms, green onion, Parmesan cheese, Asiago Cheese, and sauerkraut in a large bowl until combined. Add the olive oil, lemon juice, garlic, basil, and pepper; toss until coated. Pour this mixture into a 2 quart baking dish and sprinkle with bread crumbs.
Bake in preheated oven until the Brussels sprouts are tender, about 45 minutes.
