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In a large pot over medium-high heat, melt 2 tablespoons of butter. Sauté onion, celery, carrots, and garlic until tender.
Add chicken broth and potatoes to the pot. Bring to a boil, then cover and cook for 15 minutes, or until potatoes are tender.
Add broccoli and cauliflower to the pot and cook for 5 minutes, or until tender.
In a separate saucepan, melt the remaining butter over medium-high heat. Add flour and cook, whisking constantly, for 3-4 minutes to create a roux.
Gradually whisk in half and half until smooth.
Pour the milk mixture into the soup pot and stir. Add cheese and stir until melted. Simmer until thickened. Season with salt and pepper to taste.