Heat the olive oil in a large saucepan over medium heat. Add the celery, onion, 1/4 teaspoon of the salt and the pepper. Cook the vegetables until just tender, about 4 minutes. Add 2 cups broth and the artichoke hearts and bring to a boil. Cover, reduce the heat to medium-low and simmer until the artichoke hearts are tender, about 12 minutes.
Puree the soup, 1 cup at a time, in a blender until very smooth, adding the spinach and chopped mint with the last cup. Return the puree to the same saucepan. Mix in the remaining 1/4 teaspoon salt. Warm over low heat, thinning with the additional broth by 1/4 cupfuls if the soup is too thick. Ladle the soup into bowls. Squeeze a lemon wedge over each bowl of soup before serving.