Cook the rice according to package directions. Let stand, uncovered, until cooled.
Meanwhile, heat the oil in a medium-sized heavy skillet over medium heat and add the walnuts. Stir-fry just until fragrant, about 20 seconds. Sprinkle with the tamari or soy sauce and stir-fry, adjusting the heat so that the tamari does not burn, until the walnuts are coated with the thickened tamari, about 1 minute.
Place the broccoli in a vegetable steamer set over gently boiling water and steam covered, until tender, about 4 minutes. Lift the steamer from the saucepan, and then rinse the broccoli with cool water. Drain and cool.
To make the dressing: In a large bowl whisk together the oil, lemon juice, tamari or soy sauce, ginger, lemon zest, and garlic.
Just before serving add the cooked rice, tamari walnuts, broccoli and scallions. Toss to blend.
Note: If making the salad ahead, set aside the broccoli and tamari walnuts and add just before serving.