Cut potatoes and carrots into bite sized pieces and place in a Dutch oven. Add just enough water to cover and bring to a boil over medium-high heat, reduce heat and cook until potatoes and carrots are very tender, about 10-15 minutes. Drain, reserving 1 cup of the potatoes; return the remaining potatoes and the carrots to the pot. In a small bowl mash the potatoes with a fork and return them to the pot.

Stir in the chicken broth, salt, and pepper; heat to a boiling. Add broccoli and cook for 4 minutes stirring occasionally; reduce heat to medium.

Whisk the milk and flour together in a small bowl and add it to the pot. Bring to a boil over medium heat and cook until it thickens. Stir in cheese until melted. Add sour cream and green onions and cook until heated through.

Garnish with additional cheese and green onions, if desired.