In a large soup pot over medium-high heat, heat the oil and add the carrots, onion, celery and red pepper. Cook and stir for 3-4 minutes, until crisp-tender.
Add the garlic, salt, Italian seasoning, pepper and paprika. Cook and stir 1 minute.
Add the broth and deglaze the pot (scraping the bottom with a wooden spoon) if necessary.
Add the potatoes and bring back up to a boil. Reduce heat, cover and cook until potatoes are tender, about 5-6 minutes.
Stir in the corn, peas and green beans and heat through.
Meanwhile, in a large glass measuring cup or medium bowl, melt the butter.
Whisk in the flour until completely combined, then whisk in the milk.
Microwave on high for 1 minute, then whisk thoroughly. Repeat, using less time as needed as the mixture becomes thick and bubbles up. Keep an eye on it, as it does bubble up quickly in the end! Continue cooking and stirring until mixture is very thick.
Stir the thickened milk into the soup along with the cheese and stir until smooth.
Taste, adjust seasonings as desired and serve.
