Melt butter in a pot and cook onion until soft, then add garlic and cook briefly to build a flavorful base without burning.
Add broccoli and carrot, cooking them slightly to release sweetness and ensure even texture throughout the soup.
Stir in flour and cook briefly to create a roux, which will help thicken the soup evenly as liquid is added.
Gradually pour in broth, milk, and cream while stirring to create a smooth mixture, then simmer until vegetables are tender.
Blend part of the soup if desired to achieve a creamy consistency while still retaining some texture.
Lower heat and stir in cheese gradually until melted, then season and adjust flavor before serving.