1. In a large pot or Dutch oven, melt butter and saute onions for 4-5 minutes over medium heat.
2. Whisk in all-purpose flour and cook for 3 to 4 minutes.
3. Slowly, pour in the chicken stock, then gradually add in milk, whisking constantly.
4. Add in cream and continue to whisk and cook for four minutes. Mixture will begin to thicken.
5. Add in chopped carrots and continue to simmer for five minutes.
6. Add in broccoli florets and simmer for 20 minutes.
7. Add in dry mustard and gradually add in shredded cheese, stirring gently after each addition. Continue adding cheese as the cheese melts into the soup.
8. Taste for seasoning and season with salt and pepper to taste. Some cheeses taste more salty than others and you may not need to add any salt at all.
Leftovers may be stored in the refrigerator in a covered container for up the three days.
I do not recommend freezing this recipe.