Mix 2 teaspoons soy sauce, 1 teaspoon oil, 2 teaspoons garlic, 1 tablespoon chopped cilantro, 1 tablespoon lime juice, and 1 teaspoon ginger in medium bowl. Add chicken and toss to coat. Marinate for 4 hours or overnight.
Meanwhile, bring 2 3/4 cups water to boil in heavy medium saucepan. Add rice; return to boil. Reduce heat to medium-low, cover and simmer until rice is tender and water is absorbed, about 15 minutes. Uncover; let stand until cool.
Sprinkle chicken with salt and pepper. Heat large nonstick skillet over high heat. Add chicken with marinade to skillet and sauté until golden brown and cooked through, about 4 minutes. Cool.
Whisk remaining lime juice, 1 tablespoon soy sauce, 2 tablespoons oil, 2 tablespoons garlic and 3 teaspoons ginger in large bowl. Mix in rice, chicken, mango, 1/2 cup green onions, 1/2 cup cilantro, and bell pepper. Season with salt and pepper. (Can be made 6 hours ahead. Chill. Let stand up to 2 hours at room temperature before continuing.)
Line bowl with lettuce leaves. Spoon salad into bowl. Serve with remaining green onions and cilantro as a garnish. Serves 4.