Peel the ginger and finely grate it in to a large mixing bowl. Whisk in the olive oil, vinegar, soy sauce and sesame oil. Season with some salt and pepper. Reserve 2 tablespoons of the dressing and set aside. Thinly slice the mushrooms and scallions, keeping the white and green parts of the scallions separate. Add the sliced mushrooms, scallion whites and the edamame to the dressing in the large mixing bowl. Stir to combine and then marinate at room temperature (the mushrooms will become more tender while they sit in the dressing) while you prepare the rest of the ingredients, stirring occasionally. Reserve the sliced scallion greens to use as garnish.
Place the rice in a glass mixing bowl and break it up using a fork. Place a damp paper towel directly on top of the rice and place in the microwave; cook until just warmed through and softened, about 3 minutes.
Meanwhile, remove the skin from the chicken and shred the meat in to large bite-size pieces, about 2 cups. Toss the chicken, warm rice and carrots with the mushrooms and edamame. Season with some salt and pepper.
Toss the baby arugula and reserved dressing in a large mixing bowl. Season lightly with salt and pepper. Divide the arugula among 4 large plates and top each with 2 cups of the salad mixture. Garnish with the reserved scallion greens.
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