2 ½cups (about 8 ounces)grated sharp cheddar cheese plus more for garnish
Melt the butter in a large dutch oven or stock pot over medium heat. Add the onion and cook for 3-4 minutes until softened, stirring occasionally. Add the garlic and cook for 30-45 seconds, stirring constantly, until fragrant. Whisk in the flour and cook for 1-2 minutes until light golden brown.
Slowly add the chicken broth while whisking, and continue whisking for 30-45 seconds. Add the broccoli and carrots and stir in the salt, pepper, and dry mustard. Cover and reduce heat to medium-low to low and simmer for 20 minutes, stirring occasionally, until the broccoli is tender.
Add the half-and-half then stir in the cheese until melted. Season with additional salt and pepper to taste then serve in bowls and garnish with additional cheese.