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, being careful not to burn.
03 - Sprinkle flour into the pot and cook, stirring constantly, for 2 minutes to form a smooth roux and eliminate raw flour taste.
04 - Gradually whisk in whole milk and vegetable broth until fully incorporated and smooth, ensuring no lumps remain.
05 - Add broccoli florets and grated carrot. Bring to a gentle simmer and cook for 15-20 minutes until vegetables are completely tender.
06 - Use an immersion blender to puree to desired consistency, leaving slightly chunky or completely smooth. Alternatively, blend in batches using a countertop blender and return to pot.
07 - Reduce heat to low. Stir in heavy cream and shredded cheddar cheese, stirring constantly until cheese is fully melted and soup is creamy. Do not allow to boil.
08 - Season with salt, pepper, and nutmeg. Taste and adjust seasoning as needed. Serve hot, garnished with additional cheddar cheese and crusty bread.