01 - Melt the butter in a large pot over medium heat. Add the chopped onion and diced carrots; sauté for 5 minutes until softened and fragrant.
02 - Stir in the minced garlic and cook for 1 minute until fragrant, being careful not to brown the garlic.
03 - Sprinkle the flour evenly over the vegetables, stirring constantly for 2 minutes to form a smooth roux and cook out the raw flour taste.
04 - Gradually whisk in the vegetable broth, ensuring no lumps remain. Continue whisking until the mixture is smooth and slightly thickened.
05 - Add the broccoli florets, dried thyme, ground nutmeg, salt, and pepper. Simmer for 15 minutes until the broccoli is fork-tender.
06 - Reduce heat to low. Stir in the whole milk and heavy cream; heat gently without boiling to prevent separation.
07 - Using an immersion blender (or transferring to a countertop blender in batches), blend until desired consistency. For a chunkier texture, blend only partially.
08 - Add the grated sharp cheddar cheese, stirring constantly until fully melted and incorporated. Adjust salt and pepper seasoning to taste.
09 - Ladle hot soup into bowls and garnish with additional grated cheddar cheese if desired. Serve immediately.