Set a large soup pot over medium heat and melt butter.
Add the onions, carrots, and celery. Cook for 3 minutes, stirring frequently, until tender.
Add the garlic and continue cooking for an additional 30 seconds.
Add the broth and cauliflower. Bring to a gentle boil, then reduce to low heat and simmer for 3 minutes until cauliflower is fork tender.
In a small bowl mix flour and water, then pour the mixture into the pot. Continue simmering for 5 minutes.
Add the milk and broccoli. Simmer for a few additional minutes until chowder is heated through, and the broccoli is tender.
Stir in the cheese. When the cheese is melted, adjust seasoning to taste, and serve hot.