Whisk in the flour, then slowly add the broth, whisking between each addition to prevent lumps. Stir in the half and half, broccoli, and carrots and bring to a boil at medium high heat. Stir frequently and watch carefully! This soup likes to foam up and out of the pot when it first reaches a boil.
Once it begins boiling reduce the heat to medium and let it simmer, stirring occasionally, until the veggies are soft, about 15 minutes.
When the veggies are soft, remove from heat and stir in the cheese until melted and smooth. Serve with a batch of hot No Knead Yeast Rolls.