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Melt butter in a medium saucepan over medium heat.
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Add shallots and cook until soft, about 5 minutes.
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Stir in flour until dissolved.
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Warm milk in a microwave safe cup bowl for about 30-45 seconds until just warm.
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Gradually stir in milk about a 1/4 cup at a time until liquid thickens slightly.
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Stir in broth and cook for about 5 minutes until heated through.
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Stir in cheese until melted.
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Reserve 1 1/2 cups broccoli and 1/2 cup carrots and add the remaining broccoli and carrots into soup.
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Carefully pour mixture in a blender and and pulse a few times until mixture is creamy or desired consistency.
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Pour mixture back into saucepan. Stir in reserved broccoli and carrots. Season with salt and pepper and bring to a simmer for about 5 minutes, stirring occasionally.
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Ladle into soup bowls and serve warm.