Melt butter in a large saucepan over high heat. Cook onion, garlic, curry paste, cayenne, and salt until fragrant, stirring often, about 2 minutes. Add carrots and broth and cook, covered, until carrots are very tender, about 12 minutes.
Purée soup until very smooth, using a blender and working in batches. Stir in coconut milk and heat until hot. Mix yogurt and cilantro in a small bowl. Serve bowls of soup with a dollop of yogurt.
*Find in grocery stores' Asian-foods aisle.
Note: Nutritional analysis is per 1 1/4-cup serving.