Preheat the oven to 350 degrees F. Butter a 3-quart casserole dish.
Heat 1 tablespoon butter in a large Dutch oven or heavy pot over medium heat. Add the sausage and cook until golden brown, about 6 minutes. Add the leeks and squash and season with 1 teaspoon each salt and pepper; cook until the leeks are soft, about 3 minutes. Add the kale, toss and cover until the kale wilts, about 4 minutes. Add the focaccia cubes and the remaining 3 tablespoons butter and toss until the butter melts.
Whisk the egg and chicken broth in a bowl until smooth; stir in the diced parmesan and 1 teaspoon salt. Pour the broth mixture over the bread mixture; cook until the liquid is absorbed by the bread, about 1 minute.
Transfer the bread mixture to the prepared casserole dish. Scatter the shredded cheese evenly on top and bake until golden and cooked through, about 40 minutes. Set aside for 5 minutes before serving.