Melt the butter in a large pot or Dutch oven over medium heat.
Sauté the chopped onion for about 5 minutes, until it's soft.
Add the minced garlic and grated carrots to the pot.
Cook for another 2 minutes.
Sprinkle the flour over the vegetables.
Cook, stirring constantly, for about 1 minute or until the flour starts to turn golden.
Slowly pour in the chicken broth while whisking to prevent lumps.
Add the bay leaves and nutmeg.
Stir in the half-and-half, then add the broccoli florets.
Bring the mixture to a boil, then reduce the heat and let it simmer for 15-20 minutes.
Remove the bay leaves.
Gradually stir in the shredded cheddar cheese until it melts completely.
Season the soup with salt and pepper to taste.
Serve hot.