Preheat the oven to 350°F.
In a large skillet, heat olive oil over medium heat. Add chopped onion and garlic, sautéing until the onion is translucent, about 3-4 minutes.
Add broccoli, cauliflower, carrots, and peas to the skillet. Season with salt, black pepper, thyme, and paprika. Cook for an additional 5-7 minutes, stirring occasionally, until the vegetables are tender but still crisp.
In a large bowl, combine the sautéed vegetables with sour cream, cheddar cheese, and half of the Parmesan cheese. Mix well until all ingredients are evenly coated.
Transfer the vegetable mixture to a greased 9x13 inch baking dish. Sprinkle the remaining Parmesan cheese and breadcrumbs evenly over the top.
Bake in the preheated oven for 25-30 minutes, or until the casserole is bubbly and the top is golden brown.
Let the casserole cool for about 5 minutes before serving.