Melt the butter in a large pot or Dutch oven over medium heat. Add the chopped onion and cook for 5-7 minutes, stirring occasionally, until the onion is soft and translucent.
Add the minced garlic and cook for one more minute until fragrant. Sprinkle the flour over the onion and garlic mixture and cook, stirring constantly, for two minutes to form a pale golden roux.
Slowly pour in the chicken broth while whisking continuously to prevent lumps from forming. Whisk in the milk, dry mustard, and nutmeg. Bring the mixture to a gentle simmer, stirring occasionally.
Add the broccoli florets, diced broccoli stem, and grated carrot to the pot. Simmer for 15-20 minutes, uncovered, stirring occasionally, until the broccoli is fork-tender.
Reduce the heat to low. Gradually add the grated cheddar cheese by the handful, stirring until each addition is fully melted and incorporated before adding the next. Season the soup with salt and pepper to taste. Serve immediately.
