Heat 1/4 cup olive oil in a large stockpot over medium heat; saute onion, carrots, celery, and garlic until lightly browned, about 10 minutes. Add zucchini and cook until slightly softened, 3 to 4 minutes. Stir vegetable stock, cannellini beans, chopped tomatoes, chicken stock, and pesto into vegetable mixture; bring to a boil. Reduce heat to medium-low and simmer until vegetables are tender, 30 to 35 minutes. Season with salt and pepper.

Heat 1 tablespoon olive oil in a large skillet or frying pan over medium heat; saute spinach until wilted, about 5 minutes. Spoon sauteed spinach into each serving bowl; ladle soup over spinach.