In a large pot, heat 1 tbsp. EVOO over medium. Add the potatoes and onion and cook, stirring, until softened, 6 minutes; season with salt and pepper. Add the stock and simmer 5 minutes. Add the green beans and simmer until the potatoes are tender, about 5 minutes more. Off heat, stir in the spinach to wilt.
Meanwhile, preheat the broiler. Place the sliced ciabatta on a baking sheet. Drizzle with EVOO and sprinkle with the Parmesan; season. Broil until golden, about 3 minutes.
Divide the soup among bowls; top each with 1 tbsp. pesto and drizzle with EVOO. Serve with the Parmesan toasts.