01 - Melt butter in a large pot over medium heat. Add diced onion and carrot, sauté until softened, about 4 minutes.
02 - Stir in minced garlic and cook for 1 minute until fragrant.
03 - Sprinkle flour over vegetables and cook, stirring constantly, for 2 minutes to form a roux.
04 - Gradually whisk in vegetable broth, ensuring a smooth mixture without lumps.
05 - Add chopped broccoli, bring to a simmer, and cook uncovered for 10 to 12 minutes until tender.
06 - Reduce heat to low and puree soup using an immersion blender until smooth, leaving some texture if preferred.
07 - Stir in whole milk and heavy cream. Warm gently without boiling.
08 - Gradually add grated cheddar cheese while stirring until fully melted and smooth. Season with salt, black pepper, and nutmeg if desired.
09 - Taste and adjust seasoning if needed. Serve hot, optionally garnished with extra cheddar or fresh herbs.