Preheat oven to 400. Place potato, carrots, peas, broccoli, corn, and squash in a medium pot and fill it with enough water to cover the vegetables plus one inch. Lighly salt the water, bring it to a boil. Then allow the vegetables to continue boiling for 4 minutes. Drain and rinse with cold water and set them aside.
Melt butter over medium heat in a large, deep skillet or Dutch oven. Add the onion and mushrooms and saute them until soft, about 4 minutes. Stir in the flour and cook the mixture, stirring nonstop, another 30 seconds. Add the vegetable broth, milk, salt, pepper, thyme, and celery seed and stir until sauce thickens, about 4 minutes.
Add vegetables to the sauce and stir well. Add more salt and pepper if desired, and stir in Parmesan. Bring filling to a summer. Thin it with water if it becomes thicker than potato soup. Transfer the filling to a casserole dish.
Make the corn bread topping. In a medium bowl, whisk together cornmeal, sugar, baking powder, and salt. Add milk, egg, and oil and stir until evenly blended. Spread the batter over the filling in the casserole dish. Bake until the top is golden brown and sides are bubbly, about 25 minutes.