Place the beets into a medium pot and cover with water. Bring to a boil. Turn the heat to a simmer for 45-60 minutes, until tender. Remove from heat and chill the beets and their cooking liquid.
Roughly chop 3 beets and add 2 cups of the cold cooking liquid to a blender. Add half of the chopped red onion, garlic cloves, 1/2 English cucumber, salt, pepper, sherry vinegar, and 2/3 of the fresh dill.
Blend until very smooth. Place in the refrigerator until ready to serve.
Prepare the toppings by dicing the remaining beet, cucumber, and avocado and chopping the remaining dill.
Pour chilled gazpacho soup into bowls, top with the garnishes, and enjoy!