01 - Melt butter in a large pot over medium heat. Add chopped onion and diced carrots; sauté until softened, about 5 minutes.
02 - Stir in minced garlic and cook for 1 minute until fragrant.
03 - Sprinkle all-purpose flour over vegetable mixture and stir constantly for 2 minutes to form a roux.
04 - Gradually whisk in vegetable broth, stirring continuously to prevent lumps.
05 - Add broccoli florets, kosher salt, black pepper, and nutmeg if using. Bring to a gentle boil, then reduce heat and simmer until broccoli is tender, approximately 12 to 15 minutes.
06 - Use an immersion blender or regular blender to puree soup until smooth, or leave some texture as preferred.
07 - Stir in whole milk and heavy cream, heating gently over low heat without boiling.
08 - Gradually add grated sharp cheddar cheese, stirring continuously until fully melted and smooth. Adjust seasoning to taste.
09 - Preheat oven to 375°F (190°C). Arrange bread slices on a baking sheet and bake for 6 to 8 minutes until golden and crisp.
10 - Ladle soup into bowls, garnish with additional cheddar if desired, and serve alongside the toasted crusty bread.