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Melt butter in a pot or Dutch oven over medium heat. Add celery and onion and cook until softened, about 5 minutes.
Stir the flour into the vegetables and combine until flour is dissolved into a paste. Do not allow to burn.
Pour the broth into the vegetable mixture and stir while liquid becomes thick. Bring to a boil.
Add broccoli to boiling water and cook until bright green and soft, about 3-5 minutes.
Remove the pot from the heat and cool slightly - cool enough that the broth is no longer boiling hot, about 3 minutes. Using an immersion blender, blend the soup until about half the broccoli is broken apart and the vegetables are well blended, or use a regular blender to blend about half of the soup and return to the pot.
Heat the blended soup over low heat and stir in the carrot. Add the cheddar ¼ cup at a time. Stir to melt each addition of cheddar before adding more. Continue stirring until cheddar is completely combined, then remove from the heat. Season to taste.
Top with red pepper flakes to serve.