In a medium pot, heat olive oil over medium heat. Add onion and sauté until translucent, about 5 minutes.
Add flour and stir. Let cook for a few minutes until golden, about 5 minutes. Whisk in half and half and milk slowly until smooth. For a lower fat option, replace the half and half with milk, if desired.
Add chicken broth and bring to a simmer. Cook over low heat until thickened.
Add broccoli and carrot to the pot and cook until broccoli is soft, about 15-20 minutes.
Puree the soup in a blender on low, allowing some larger flecks of carrot and broccoli to remain. Return to pot. Alternatively, you could keep the soup in the pot and blend with an immersion blender.
Add cheese and stir until melted. Serve immediately.
