Preheat your oven to 375°F (190°C).
In a large skillet, heat a tablespoon of olive oil over medium heat. Add the onion, carrots, celery, and bell pepper and cook until softened about 5-7 minutes.
Stir in the broccoli florets, corn kernels, and peas. Cook for a few more minutes until the vegetables are tender-crisp.
In a separate bowl, whisk together the cream of mushroom soup, milk, Parmesan cheese, thyme, salt, and pepper.
Pour the cooked vegetables into a 9x13-inch baking dish. Top with the rice or pasta, then pour over the creamy sauce.
Bake in the preheated oven for 25-30 minutes, or until the casserole is bubbly and the cheese is golden brown.
