Melt the butter in a Dutch oven over medium heat. Add the onion and saute until soft, about 4-5 minutes.
Sprinkle the flour over the onions. Cook and stir over medium heat for 1-2 minutes to cook out the raw flour taste.
Whisk in the half-and-half and chicken broth. Bring to a boil then reduce to a simmer.
Add the broccoli and carrots. Cook over medium-low heat for 20-25 minutes or until the veggies are tender.
For a smoother soup, blend about 1 cup of the soup with an immersion blender and stir back in.
Remove the pot from the heat and stir in the cheese (adding cheese with the heat on can make it turn grainy). Stir in the nutmeg if desired. Add salt and pepper to taste.
Gluten-free: Skin the flour roux and thicken in the end with a cornstarch slurry (2 tablespoons cornstarch whisked with a little water).Storage: Keeps in the fridge 3-4 days; reheat gently on the stove top since cheese-based soups can separate in the microwave.
Serving: 1/6 of the recipe, Calories: 413kcal, Carbohydrates: 19g, Protein: 17g, Fat: 31g, Saturated Fat: 18g, Polyunsaturated Fat: 10g, Trans Fat: 1g, Cholesterol: 90mg, Sodium: 610mg, Fiber: 3g, Sugar: 7g