Sprinkle the sliced cucumber with salt and place in a single layer on a parchment-lined sheet. Set aside for 30 minutes. Pat dry with paper towels. Whisk together sugar and vinegar in a bowl and add the cucumber slices; toss to coat.
Cook the quinoa according to package instructions.
Place the beets in a large pot, cover with water, bring to a boil and cook until tender, about 30 minutes; drain and return to the pot.
Whisk together all the dressing ingredients in a small bowl and taste and adjust seasoning.
Divide the quinoa between bowls and top with cucumber, beets, radishes, carrots, edamame and sprouts. Drizzle with the dressing and serve.