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Melt butter in a large stockpot or Dutch oven over medium heat. Add garlic, onion and celery, and cook, stirring occasionally, until translucent, about 3-4 minutes.
Whisk in flour until lightly browned, about 1 minute. Gradually whisk in vegetable broth, milk and heavy cream, and cook, whisking constantly, until incorporated, about 1-2 minutes. Stir in broccoli and carrot.
Bring to a boil; reduce heat and simmer until broccoli is tender, about 6-8 minutes. Puree with an immersion blender until desired consistency is reached.
Stir in cheese, a handful at a time, until smooth, about 1-2 minutes; season with salt and pepper, to taste.
Serve immediately.