01 - Melt butter in a large pot over medium heat. Add chopped onion and cook until soft and translucent, about 4-5 minutes.
02 - Stir in minced garlic and cook for 1 minute until fragrant.
03 - Sprinkle flour into the pot and cook, stirring constantly, for 2 minutes to form a smooth roux.
04 - Gradually whisk in whole milk and vegetable broth until completely smooth and no lumps remain.
05 - Add broccoli florets and grated carrot. Bring to a simmer and cook for 15-20 minutes until vegetables are tender.
06 - Use an immersion blender to puree soup to desired consistency. Alternatively, blend carefully in batches using a countertop blender, then return to pot.
07 - Stir in heavy cream and shredded cheddar cheese. Cook over low heat, stirring constantly until cheese is fully melted and soup is creamy. Do not allow to boil.
08 - Season with salt, pepper, and nutmeg if using. Taste and adjust seasoning as needed. Serve hot, garnished with extra cheddar cheese and crusty bread.