In a Dutch oven or similar large, deep pot, add the oil and heat over medium-high. Add the carrots and onion. Let cook, stirring often, until the onions begin to soften, about 8 minutes.
Add the garlic, fresh ginger, salt, cumin, turmeric, ground ginger, and cayenne and let cook until fragrant, about 30 seconds.
Add the broth. Increase the heat to high and bring the soup to a boil. Reduce the heat to maintain a steady simmer. Let simmer until the carrots are tender, about 30 minutes.
Turn off the heat. Puree the soup: with an immersion blender, blend the soup until smooth and creamy. Once it's mostly pureed, add the Greek yogurt (or coconut milk) and blend again. If the soup is thicker than you would like, add more broth (or water) as needed). Alternatively, you can transfer the soup to a blender and puree it in batches, then return it to the pot—add the Greek yogurt while blending. (Be careful not to overfill your blender; hot soup will splatter!).
Stir in the vinegar. Carefully taste and season with additional salt as needed (depending on your broth you may want another pinch). Enjoy hot, topped with fresh cilantro.