Gather all of your ingredients and have them ready to go. (vegetables chopped, cheese grated, etc.)
Melt the butter in a large, heavy bottomed soup pot over medium high heat. Add in the chopped onions.
Fry the onions until they are soft and translucent. Make sure that they are as soft as you want them, as onions can stop cooking when you add another ingredients to them.
Add in the garlic to the pot and cook for 1 minute or until it is are browned and fragrant , stirring the entire time so that it doesn't burn or stick to the pan.
Next, you use the Singer method to make a roux. Sprinkle the flour over the fried vegetables. whisking and mixing into the butter, ensuring that there are no lumps ( excepting the onions and garlic, of course). Stir and cook for 3-4 minutes, allowing the flour to cook properly and thicken up with the butter.
Once the flour is cooked, very slowly pour in the chicken stock, little by little, whisking continuously, making sure that there are no lumps. Stir until the flour has dissolved into the the broth and the broth is a smooth mixture now.
Once the chicken stock has been added and the mixture is smooth, add in the milk and heavy cream, stirring continuously.
Add and stir in the mustard powder, the cayenne and the nutmeg if desired.
Turn down the heat to medium-low.
Add in the broccoli and carrots and simmer on lower heat until the broccoli florets are soft.
Stir in the shredded cheese and mix until the cheese has melted throughout the broth and is creamy and smooth.
If desired, you can thicken this soup even more by pureeing 1/3 of the soup, then returning it to the pot. I also use an immersion blender and simply blend it in the pot for 5-10 seconds, making sure not to blend the entire soup, leaving some broccoli chunks.
Serve in bowls topped with shredded cheese, salt and pepper.
Store in an airtight container in the fridge for 3-5 days. Cream soups do not freeze well.
Notes
Do not boil the soup, just heat gradually and let simmer on low. Simmering will be steamy, not rolling with bubbles!
If you are using a weak flavored chicken or vegetable broth, you can add powdered bouillon to make it stronger and enhance the soup, this is where a lot of the flavor comes from so don't skimp.
If the soup is too thick, thin it out with additional broth. Wait until the very end however to do this, or you might thin it too much!