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Melt butter in a stock pot over medium heat. Add the carrots and the onion and sauté until tender, 4 to 5 minutes. Whisk in the flour and continue to stir for about 5 minutes. Slowly add the half and half and continue to stir until smooth. Pour in the vegetable broth and stir until well blended. Add the salt, pepper and the garlic.
Reduce the temperature to medium low and simmer until mixture thickens, about 8 to 10 minutes.
Add the broccoli and simmer until tender, about 15 minutes. Add the cheese and cook until cheese is melted, 4 to 5 minutes.