In a large pot, melt the butter over medium heat. Add the onion and garlic, sauté until softened, about 5 minutes.
Sprinkle the flour over the onion mixture and stir constantly for 1 minute to cook the flour.
Gradually whisk in the broth and milk, stirring constantly to prevent lumps.
Add the broccoli, carrots, nutmeg, salt, and pepper. Bring to a simmer and cook until the vegetables are tender, about 15-20 minutes.
Use an immersion blender to partially puree the soup, leaving some chunks for texture. Alternatively, transfer half the soup to a blender, puree, and return to the pot.
Stir in the cheddar cheese until melted. Add the cream if using, and adjust seasoning to taste.
Serve hot, garnished with extra cheese and chopped chives if desired.