Wash your hands well with soap and warm water for at least 20 seconds.
Peel and cut onion into small pieces.
Heat butter in a medium to large pot over medium-high heat, add onion and carrots and cook until vegetables are tender, about 3 to 5 minutes.
Add broth and broccoli to the pot and cook until broccoli is tender, about 5 minutes.
In a small bowl, add milk and flour and mix until smooth. Add the milk mixture to the pot, bring to a boil, and continue to stir until soup thickens slightly, about 3 to 5 minutes.
Turn heat to low and add cheese, pepper, and salt to the pot and stir until cheese is melted.