01 - In a large pot over medium heat, melt the butter. Add the onion and cook until soft and translucent, about 4-5 minutes.
02 - Stir in the garlic and cook for 1 minute until fragrant.
03 - Sprinkle in the flour and cook, stirring constantly, for 2 minutes to form a roux.
04 - Gradually whisk in the milk and vegetable broth until smooth.
05 - Add the broccoli and grated carrot. Bring to a simmer and cook for 15-20 minutes, until the vegetables are tender.
06 - Use an immersion blender to puree the soup to your desired consistency. Alternatively, carefully blend in batches using a countertop blender, then return to the pot.
07 - Stir in the heavy cream and shredded cheddar cheese. Cook over low heat, stirring until the cheese is fully melted and the soup is creamy. Do not let it boil.
08 - Season with salt, pepper, and nutmeg. Taste and adjust seasoning as needed. Serve hot, garnished with extra cheddar and crusty bread if desired.