Wash and dry the head of cauliflower.
Cut the cauliflower into bite-size pieces. Prepare the carrots and onion.
Add 2 tablespoons butter, cauliflower, sliced carrots, and chopped onion to a large soup pot or Dutch oven.
Sautee the vegetables over medium heat until slightly tender 8 to 10 minutes. Stir frequently so that the vegetables don't brown.
Add the remaining butter and flour and stir in until it is combined with the vegetables.
Stir in the chicken broth. Add the milk and stir in.
Stir in the cheese. Continue to stir until the cheese is melted.
Cook until the vegetables are tender about 20 to 30 Minutes.