Dice up your onions and garlic very small. Then chop up your broccoli, making sure the broccoli florets are small and the stems are in very small pieces.
Melt butter in small saucepan then saute onion in it. When onion are golden, stir in the garlic and broccoli and cook for 3-4 minutes, then set aside.
In a large pot, melt the rest of the butter and add in flour over medium heat. Heat for 4-5 minutes, stirring constantly with a whisk.
Add the half and half and chicken stock and simmer for 15-20 minutes until it has thickened.
Add the onion, garlic and broccoli mixture, as well as the carrots. Cook over low heat for another 25-30 minutes until all veggies are tender.
Add lots of salt and pepper and pour half the mixture into a blender to puree the soup. Return the soup to the pot and add the grated cheese, then stir well.
Keep soup over low heat and add in the nutmeg. Enjoy!