In a large non-stick stock pot, melt the butter over medium heat. Add the diced onion and sauté for 4-5 minutes. Add the garlic and cook an additional minute.
Sprinkle in the flour and cook for an additional 2 minutes, stirring constantly. (The mixture will be thick.)
Whisk in the chicken stock until the flour has dissolved.
Stir in the broccoli florets, shredded carrot, salt, pepper, mustard powder and nutmeg and allow the mixture to simmer over medium-high heat for about 15-20 minutes.
Stir in the half and half and the grated cheese. Bring to a simmer for 2-3 minutes (still on medium-high heat) until everything is melted and well combined.
Serve immediately with crusty bread and garnish with extra cheese if desired!