Heat the oil and butter in a large pot over medium heat. Once the butter has melted, add the onion, celery, and carrots. Sauté for 5 minutes or until the onion starts to turn translucent.
Add the garlic, thyme, paprika, and cayenne pepper. Saute for another 1-2 minutes or until fragrant.
Stir in the potatoes, broth, salt, and black pepper. Increase the heat to medium-high and bring the soup to a simmer. Reduce to medium and cook for 10 minutes.
Stir in the corn and zucchini. Return the soup to a simmer and cook for 10 minutes.
Reduce the heat to low. Whisk the cornstarch into the half-and-half. Stir the half-and-half and Parmesan cheese into the soup and heat for 2-3 minutes or until heated through.
Carefully transfer 2-3 cups (depending on your preference – see notes below) of the soup to a blender and blend with the lid vented until smooth or blend with an immersion blender until the soup reaches your preferred consistency.
Stir the blended soup into the pot with the remaining soup. Season to taste with salt and pepper and enjoy hot with your toppings of choice.