In a large stockpot or Dutch oven, heat coconut oil over medium heat. Add leek, garlic, and ginger; cook until softened, 2 minutes. Add sweet potatoes, persimmons, salt, curry powder, ½ teaspoon paprika, and red pepper; cook, stirring occasionally, until aromatic, 2 minutes.

Add broth; bring to a boil. Reduce heat, cover, and cook until sweet potatoes and persimmons are tender, about 20 minutes.

Working in batches, transfer mixture to the container of a blender; process until smooth. Return soup to pan; cook over medium heat. Add coconut milk; cook stirring occasionally, until heated through. Do not boil.

In a small bowl, combine olive oil and remaining ¼ teaspoon paprika. Drizzle oil over servings of soup. Top with pistachios.