Melt the butter in a large pot over medium heat. Add the onions and carrots. Season with salt, pepper, and nutmeg. Sauté for three to four minutes or until the onions are tender.
3 tablespoons unsalted butter, 2 cups shredded carrot, 1 small yellow onion, 1 teaspoon kosher salt, 1/2 teaspoon black pepper, 1/8 tsp nutmeg
Whisk in the flour, and cook for about 30 seconds
1/3 cup all purpose flour
Very slowly pour in the chicken stock. Add about two tablespoons at a time, fully whisking it in and combining it before adding more. As you add more, you will be able to add it more quickly.
6 cups chicken broth
Stir in the broccoli, bring to a simmer, and cook for 15 minutes or until the broccoli is soft and tender.
4 cups broccoli florets
Optional: Remove some of the soup and blend it for a less chunky soup.
Stir in the half and half and allow the soup to warm through.
1 1/2 cups half and half
Stir in the cheddar cheese until it is fully melted and combined into the soup.
2 cups mild cheddar cheese