In a large pot, heat a splash of oil over medium heat. Add the diced onion, carrots, and garlic. Sauté for about 5 minutes until softened.

Add the broccoli florets and sauté for another 3 minutes.

Pour in the vegetable broth and bring to a boil. Reduce heat and let it simmer for 15 minutes until the broccoli is tender.

Using an immersion blender, blend the soup until smooth. Alternatively, transfer to a blender in batches.

Return the blended soup to the pot. Stir in the milk, shredded cheddar cheese, and cooked chicken. Cook over low heat until the cheese is melted and the soup is heated through.

Season with salt, black pepper, and paprika. Serve hot, garnished with additional cheese if desired.