Sauté onion in butter. Set aside.
Cook melted butter and flour using a whisk over medium heat for 3-5 minutes or until you see a noticeable golden brown color in your pan.
Slowly add in the chicken stock and whisk again to combine. Simmer contents covered (stirring occasionally) for 20 minutes on medium heat.
Add the broccoli, carrots and the sauteed onions, into the pot. Stir, add the milk and half & half. Cook covered over low heat 20-25 minutes but do not bring to a boil. This may cause the milk to curdle.
Add salt, pepper & nutmeg. Note: you can purée half of your soup in a blender or with a handheld immersion blender if you choose however it isn't a requirement. Continue to cook the soup on low heat and slowly add the grated cheese a handful at a time and stir to avoid clumps. Once all the cheese has been added is melted, remove from heat, and serve immediately. Suggestion: serve with crusty bread or in a breadbowl.
Refrigerate leftovers after they have cooled and store in a airtight container up to 5 days in the refrigerator.