Ladle about 1 cup of vegetables and about half as much broth into a blender. Summer squash and other watery vegetables might not need any added broth. Cover, but remove the center cap or keep the lid slightly cracked with the opening away from you. Cover the lid with a folded dishtowel.
Purée in batches. Blend in pulses or on low speed at first, and then continuously until you have a completely smooth purée, adding more broth only if needed. With the blender on, carefully ladle in more vegetables and broth through the feeder hole, keeping the ratio about the same, until the blender is half full.
Strain the purée through a fine sieve set over a bowl, pressing it through with a rubber spatula. Purée and strain the remaining vegetables, including any spices. Thin the soup, if necessary, so it has the consistency of heavy cream (you may not use all the broth). Continue as the recipe directs.