Sauté Aromatics: In a large pot or Dutch oven, melt the butter over medium heat. Sauté the onions, cumin, and garlic until the onions are soft and translucent, about 5–7 minutes.
Boil: Add the carrots, sweet potato, and chicken broth to the pot. Bring the mixture to a boil, then reduce heat and simmer until the carrots and sweet potatoes are fork-tender and cooked through (about 15-20 minutes).
Purée: Carefully remove the pot from the heat. Using an immersion blender, purée the soup until it is completely smooth and creamy. (Alternatively, let the soup cool slightly and blend in batches using a stand blender.)
Finish & Season: Return the puréed soup to the pot over low heat. Stir in the coconut milk, lime juice, and brown sugar. Heat through, then add salt and pepper to taste.
Serve: Ladle into bowls and enjoy your perfect homemade creamy carrot soup!