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In a large pot, bring the stock, potatoes, carrots, onions, and celery to a boil.
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Add salt and pepper.
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Cover and reduce heat to medium. Cook for 8-10 minutes or until the vegetables are tender.
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Add the broccoli. Cook for 2 minutes.
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Add the cream cheese and stir frequently as to facilitate melting.
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Slowly add both the cheddar cheeses.
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The soup will quickly begin to thicken as the cheese melts.
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Add the milk, a half of a cup at a time, stirring constantly until the soup has reached the desired thickness.
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Taste, and add any additional salt and pepper as needed.
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If an even thicker soup is desired, remove 1 cup of the hot liquid and place it into a bowl.
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Add the cornstarch, (no more than 2 tablespoons) a small amount at a time.
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Whisk briskly to make sure the cornstarch does not clump.
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Pour the thickened broth back into the pot and stir well, making sure it is fully incorporated into the soup.
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Cook on low for 5 minutes. Serve and garnish with green onions.