1. Heat oil in a large pot over medium. Add leeks, carrots, and celery; cook, stirring, until tender, 6 to 8 minutes. Add broth and peas; bring to a boil, then reduce to a simmer and cook, partially covered, until peas are creamy, 50 to 55 minutes. Stir in spinach; cook until just wilted, about 5 minutes. Let cool slightly.

2. Working in batches, puree soup in a blender until smooth. Return to pot; season with salt and pepper. (If too thick, add water, 1 tablespoon at a time.) Ladle soup into bowls, drizzle with yogurt and oil, top with barley, and serve.