Chop broccoli into florets. You can include some to the stem, but avoid the thick parts.
In a large pot over medium-high heat, melt half the butter. Once melted, add the onions and celery. Saute for 5 minutes.
Add broccoli and broth. Cover and let cook for 10 minutes.
Remove from heat. Use an immersion blender to puree the soup until smooth (if you don’t have an immersion blender, you can use a blender. NOTE: when blending hot liquids in a blender, remove the small vent on the top of the lid and cover with a towel when blending).
Once smooth, return to low heat.
In a small pot over medium-heat, bring milk to a boil. Reduce heat to a simmer, add flour one tablespoon at a time mixing thoroughly.
Add milk to broccoli and mix well. Simmer for 10 minutes.
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