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Melt butter in a large stock pot. Meanwhile, using a food processor, grate onion, carrots, and garlic.
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Add grated vegetables to the butter, then season to taste with salt and pepper. Cook over medium-high heat for 3-5 minutes or until vegetables are tender.
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Add flour and mix together well, then stir for about a minute.
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Add vegetable broth and half and half, then whisk until smooth. Stir in bay leaves, nutmeg, and 1 cup of reserved small broccoli florets. Bring mixture to a boil, then cover, and simmer for 15 minutes. Meanwhile, grate the remaining broccoli using the grater plate of a food processor.
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Remove the bay leaves. Add grated broccoli, cover, and cook until tender, about 10 minutes.
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Gradually add grated cheese and whisk until well combined.
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Serve immediately.