Preheat the oven to 375°F. Grease a 9x13-inch baking dish with cooking spray or olive oil.
In a large bowl, combine the sliced zucchinis, sliced potatoes, chopped onion, and minced garlic. Add the thyme, salt, and black pepper, and mix well.
In a separate bowl, combine the sour cream and half of the cheddar cheese. Stir this mixture into the vegetable mixture until everything is well coated.
Transfer the vegetable mixture into the prepared baking dish, spreading it evenly.
In a small bowl, mix the breadcrumbs with the remaining cheddar cheese, olive oil, and Parmesan cheese. Sprinkle this topping evenly over the casserole.
Cover the baking dish with aluminum foil and bake for 30 minutes.
Remove the foil and bake for an additional 15-20 minutes, or until the top is golden brown and the vegetables are tender.
Let the casserole cool for about 5 minutes before serving.