Halve cucumber lengthwise; cut one half into eighths and the other into 1/4- inch dice. Set aside. Cut asparagus spears into 2-inch lengths. Puree in a blender with 1/2 cup cold water until smooth.
Add spinach, scallions, cucumber eighths, and another 1/2 cup water. Blend until thoroughly pureed. Add avocado, mint leaves, and lemon juice; puree until smooth, adding remaining water a little at a time until soup is desired consistency. Add salt, and season with pepper. Scrape down sides of blender with a rubber spatula, and puree 5 seconds more.
Cut sorrel into fine strips, if using. Divide soup among four bowls; garnish each with diced cucumber, sorrel, and a mint sprig. Serve immediately.