Preheat oven to 350°F. Butter or spray a large casserole dish (about 3 quarts) and set it aside aside.
In a large pan or Dutch oven over medium-high heat, melt 1 tablespoon butter. Add sausage and cook until browned.
Add leeks (or onions), cubed squash, salt & pepper. Cook until leeks or onions are tender, about 4-5 minutes.
Stir in the chopped kale and cook until it begins to wilt, about 4-5 minutes.
Add the stale bread cubes and the remaining 3 tablespoons of butter, and toss together to combine. Reduce heat to low.
In a bowl (or a large glass measuring cup for easy pouring), whisk together broth, egg, cheese and a pinch of salt until combined.
Pour egg mixture over bread and cook for about a minute, or until most of the moisture is soaked up by the bread.
Pour the mixture into the prepared casserole dish and sprinkle with remaining ½ cup of Parmesan cheese.
Bake for about 40 minutes, until top is golden and mixture is cooked through. Cool for 5-10 minutes before serving