In a large bowl, combine the chicken, scallions, half the ginger, the juice from the lime, half the oil, 1 teaspoon of the salt, and half the ground pepper. Toss to combine well and set aside for 15 minutes.
Meanwhile, in a large saucepan, combine the rest of the ginger and the juiced lime halves. Add about an inch of water, place a large steamer basket (or large colander) in the pan, and bring to a boil.
Remove the chicken from the marinade and place in the steamer in a single layer. Steam until cooked through, about 5 minutes, flipping once halfway through. Remove to a plate and cover to keep warm. Reserve 3 cooked chicken-breast halves for another use.
In a large bowl, whisk together the lemon juice and the remaining oil, salt, and pepper. Add the spinach, cucumber, red onion, and chili pepper. Toss well. Serve the remaining chicken topped with the Cucumber-Spinach Salad.