In a large pot, melt butter over medium heat. Stir in onion and garlic, and cook until softened 2 to 3 minutes.
Stir in the broccoli and carrots, cooking for five more minutes.
Add flour over the vegetables and mix well until coated. Cook for 1-2 minutes to cook the raw flour taste out.
Add the chicken broth slowly while stirring to prevent lumps. This mixture is brought to a simmer.
Then stir in the milk, cooked chicken, salt, pepper, and nutmeg (if used). Simmer for 10 minutes or until the broccoli is tender.
Reduce heat to medium-low and stir in the cheddar cheese, a handful, until thoroughly melted and integrated.
Taste and adjust the seasoning as needed. Serve warm with good crusty bread or crackers.