Combine the cucumber, vinegar and sugar; mix until the cucumber is coated. Let marinate for at least 15 minutes, while you prep everything else.
Put the edamame in a colander in the sink. Cook the noodles in a saucepan of boiling water, then pour the noodles and cooking water over the frozen edamame in the colander. Let sit for 30 seconds to thaw the edamame, then rinse under water to prevent the noodles from clumping.
Heat a dry, nonstick skillet over medium heat. Add the tofu and cook until golden on the first side, 5-10 minutes, then flip and cook on the other side. Remove from the pan.
Mix all the dressing ingredients together.
Toss the noodles and edamame with the arugula, mango, scallions, basil, and nuts. Divide the salad between 3 plates and top with the dressing and tofu.