Grate the beets, cucumber, apple, and carrot in a food processor fitted with the grating blade. Transfer the grated ingredients to a soup tureen or some other kind of serving container such as a deep, steep-sided casserole dish.
Fill the container with enough water to give the soup a dense but not overly crowded consistency. Stir in the dill and scallions.
Add the juice of 1 to 1 ½ lemons (depending on how large and juicy they are), and the agave. Season gently with salt and pepper.
Cover and let the soup chill for at least 3 hours. This can be made the morning before you want to serve it for dinner, or the day before. Just before serving, taste and add more lemon juice and agave to your liking.
Serve garnished with a dollop of vegan sour cream or cashew cream, if desired.