Make the dressing: Combine all the ingredients for the dressing in a blender or food processor. Season with ¼ teaspoon salt and 1/8 teaspoon pepper. Puree until smooth and emulsified. Taste and adjust for seasoning with salt and pepper (see note). Set aside.
Make the salad: In a medium bowl, combine cucumber, avocado and half the green onions. Drizzle with enough dressing to coat. Toss gently to combine.
To serve: Arrange the cucumber and avocado mixture in a serving bowl or platter. Scatter the remaining green onions, radishes, red onions and cheese on top. Drizzle with desired amount of dressing and season to taste with salt and pepper. Serve immediately and enjoy!