Melt butter in a heavy pot over medium heat; cook and stir onions until translucent, about 10 minutes. Add chicken broth and potatoes; bring to a boil. Reduce heat to medium-low, cover pot, and simmer, about 5 minutes. Add carrots and simmer, about 10 minutes. Add zucchini and simmer until all vegetables are tender, about 15 minutes.
Combine soup, basil, salt, and pepper into a blender or food processor no more than half full. Cover and hold lid down; pulse a few times before leaving on to blend. Puree in batches until desired consistency is reached.