Place beets in a medium pot and cover with water. Bring to a boil. Turn heat down to low and simmer until fork tender all the way through, about 45- 60 minutes. Chill beets and their cooking liquid.
Once beets are cold, slip off their skins using your hands. Slice and place 3 of the 4 beets ( saving one) in a blender with 2 Cups of the cold cooking liquid ( or use ice water, or cold veggie stock). Add the half of the chopped onion( about ⅛ cup) , 2 garlic cloves, 2 sliced turkish cucumbers ( saving one) salt, pepper, vinegar and about ⅔ of the fresh dill ( saving some for garnish). Blend until very smooth. Taste and adjust salt and vinegar. Place in the refrigerator until ready to serve.
Prep the garnishes. Finely dice remaining beet, cucumber, avocado and chop the remaining dill. Pour chilled beet soup (the colder it is, the better) into bowls. Top with with the garnishes. Drizzle with a little olive oil or a swirl of yogurt or sour cream if you like. Serve immediately!