01 - Melt butter in a large pot over medium heat. Add chopped onion and diced carrots, cooking until softened, approximately 5 minutes.
02 - Incorporate minced garlic and cook for 1 minute until fragrant.
03 - Sprinkle flour over the vegetables, stirring constantly to form a roux for 2 minutes.
04 - Gradually whisk in the vegetable broth, ensuring the mixture is smooth and free of lumps.
05 - Add broccoli florets, salt, pepper, and nutmeg. Bring to a gentle boil, then reduce heat and simmer until broccoli is tender, about 12 to 15 minutes.
06 - Puree the soup using an immersion blender or in batches with a standard blender until smooth or desired texture is reached.
07 - Stir in whole milk and heavy cream, heating gently over low without boiling.
08 - Add grated cheddar cheese gradually, stirring until fully melted and integrated. Adjust seasoning as necessary.
09 - Preheat oven to 375°F. Arrange bread slices on a baking sheet and bake for 6 to 8 minutes until golden and crisp.
10 - Serve the soup hot, optionally garnished with extra cheddar, accompanied by the toasted crusty bread.