01 - Melt butter in a large pot over medium heat. Add chopped onion and diced carrots; cook until softened, about 5 minutes.
02 - Incorporate minced garlic and cook for 1 minute until fragrant.
03 - Sprinkle flour over vegetables and stir constantly for 2 minutes to form a roux.
04 - Gradually whisk in vegetable broth, ensuring no lumps develop.
05 - Add broccoli florets, salt, pepper, and optional nutmeg. Bring to a gentle boil, then reduce heat and simmer until broccoli is tender, approximately 12 to 15 minutes.
06 - Using an immersion blender or standard blender in batches, puree soup to desired consistency, leaving texture if preferred.
07 - Stir in whole milk and heavy cream; heat gently over low heat without boiling.
08 - Add grated cheddar in handfuls, stirring until fully melted and smooth. Adjust seasoning to taste.
09 - Preheat oven to 375°F. Arrange bread slices on a baking sheet and bake 6 to 8 minutes until golden and crisp.
10 - Ladle soup into bowls, garnish with extra cheddar if desired, and serve alongside toasted bread.