Melt the butter in a stock pot over medium Heat add the carrots and the onions and saute until tender 4 to 5 minutes whisk in the flour and continue to stir for about 5 minutes slowly add the half-and-half continue to stir until smooth pour in the vegetable broth and stir until well Blended and add the salt and pepper and the garlic

Reduce the temperature to medium-low and simmer until mixture thickens about 8 to 10 minutes

Add broccoli and simmer until tender about 15 minutes add the cheese and cook until cheese is melted 4 to 5 minutes