In a large pot, melt the butter over medium heat. Add the onion and garlic and cook until the onion is translucent, about 5 minutes.
Stir in the flour and cook for 1 minute.
Gradually add the chicken broth, stirring constantly, until the mixture is smooth.
Add the broccoli, carrot, salt, pepper, and nutmeg to the pot. Bring the mixture to a boil, then reduce the heat to a simmer and cook until the vegetables are tender, about 15 minutes.
Remove the pot from the heat and use an immersion blender to puree the soup until smooth. Alternatively, you can transfer the soup to a blender and puree it in batches.
Stir in the cheddar cheese and heavy cream. Cook over low heat until the cheese is melted and the soup is heated through.