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 it.
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 blended and smooth, ensuring no lumps remain.
05 - Add broccoli florets and grated carrot. Bring to a simmer and cook for 15–20 minutes until vegetables are completely tender.
06 - Use an immersion blender to puree to desired consistency (smooth or slightly chunky). Alternatively, blend carefully in batches using a countertop blender, then return to pot.
07 - Stir in heavy cream and shredded cheddar cheese over low heat. Cook, 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 and crusty bread if desired.