Heat olive oil in heavy large pot over medium heat. Add onion and sauté until soft, about 4 minutes. Add broth and next 7 ingredients. Increase heat to high and bring soup to boil. Reduce heat to medium-low, partially cover pot, and simmer until potatoes are tender, about 10 minutes.
Stir in spinach; simmer 3 minutes longer. Season soup to taste with salt and pepper. Ladle soup into 6 bowls; garnish each with 1 tablespoon pesto. Serve, passing cheese separately. Makes 6 servings.