Melt 3 tablespoons butter in a medium stock pot. Saute carrot, onion, and celery until the onions become tender. Add broccoli and broth. Cover and simmer over medium heat for 20 minutes.

Pour some of the mixture into a blender until it's half full. Let cool for 5 minutes. Cover the lid with a kitchen towel to catch splatters. Pulse a few times before pureeing until smooth. Add mixture back to pot and stir.

While the broccoli mixture is continuing to simmer. In a small bowl, whisk together milk, cream, and flour until smooth. In a small pot, over medium heat melt the remaining butter. Slowly whisk in the milk mixture. Continue to stir until thick. Add to broccoli mixture.

Stir in cheese and season to taste. Enjoy!