In a heavy small saucepan over medium heat, warm 1tablespoon of the olive oil. Add the shallot, ginger, and chile and sauté until tender, about 3 minutes. Add the tamari, brown sugar, vinegar, and 3tablespoons water and simmer until the mixture is reduced to ¾cup (6fl oz/180ml), about 6 minutes. Remove from the heat and stir in the sesame oil. Transfer half of the mixture to another small bowl.
In a large bowl, place the chicken. Add 1tablespoon of the olive oil and salt and pepper to taste and toss to coat. In another large bowl, combine the bok choy and mushrooms. Add the remaining 2tablespoons olive oil, and salt and pepper to taste, and toss to coat.
Prepare a grill for direct-heat cooking over high heat. Place the chicken on the grill rack and brush with sauce from one of the bowls. Place the vegetables on the grill rack. Cover the grill and cook the chicken until cooked through, brushing occasionally with the sauce, about 6 minutes per side. Cook the vegetables until tender-crisp and browned, about 4 minutes per side. Transfer the chicken and vegetables to a warmed platter and sprinkle with the green onions. Serve right away with the rice and remaining sauce.