Melt 1 Tbsp. butter in a large pot. Saute carrots, onion and broccoli in the butter about 5 minutes, stirring occasionally. Add broth and salt. Bring to a boil. Reduce heat and cover, simmering for 10 minutes.
While vegetables simmer: In a medium pan, melt remaining 3 Tbsp. of butter. Add flour. Cook and stir 3-5 minutes, until bubbly.
Add roux to vegetables, and bring back to a boil. Cook and stir 2 minutes.
Reduce heat to low. Add remaining ingredients and stir until cheese is melted and soup is hot.