Prepare: Chop the onion and garlic first and then start heating the oil in a large soup pot while you start chopping the the carrot and ginger. Once you add the onion and garlic to the pot, you can finish chopping the carrot and ginger and gathering the cumin, coriander, broth and coconut milk.
Saute Onion and Garlic: If you're using oil, heat it in a large soup pot over medium heat then add the onion and garlic with a pinch of salt and pepper and cook for 3-4 minutes until fragrant and soft. For oil-free soup, use water or broth instead of oil.
Add Ginger, Carrot and Spices: Add the ginger, carrot, 1 tsp cumin and 1 tsp coriander, along with another pinch of salt and pepper, and cook for 1-2 minutes, stirring constantly. If the pan starts to dry out, you can add a splash of water or broth.
Simmer Soup: Add the coconut milk and vegetable broth and let it simmer lightly, uncovered for about 20 minutes or until the carrots are fork tender and cooked through. Remove the soup from heat.
Blend Soup: Use an immersion blender in the pot or carefully transfer the soup to a blender and blend until smooth and creamy. Pour it back into the pot.
Season and Serve: Taste the soup and adjust the seasoning if needed. Serve right away with finely chopped cilantro, a drizzle of coconut milk and a squeeze of fresh lime juice.