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In large pot, melt butter.
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Add carrots, celery, garlic and onions and sauté until translucent.
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Add broccoli and then pour 1-48 ounce can of chicken stock into the pot.
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Season with salt, pepper and bay leaf.
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Cover and simmer for 20 minutes.
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Once broth is finished simmering, remove bay leaf. Take out about 1 1/2 cups for the broccoli from the soup and set aside.
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To thicken soup, mix 1 cup of cold chicken broth with 2/3 cup of flour.Bring soup to a boil and slowly pour into soup. Soup will thicken once it comes to a full boil.
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Reduce heat to a slow simmer.
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Use an emulsion blender to puree the soup in the pot. Blend until the chunks of vegetables are well pureed.
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Add the broccoli that was set aside back into the soup.
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Finish soup off by adding 2 cups of half and half and 2 cups of shredded cheddar cheese. Bring to simmer for 4 minutes and reduce to low.
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Garnish bowl with extra cheddar cheese.