Step 1In a bowl, whisk together the vinegar, tahini, oils, ginger, soy sauce, sugar and garlic. This makes about 3/4 cup dressing, which will keep up to 4 days, covered and refrigerated.
Step 1In a heavy saucepan, bring the water, barley and salt to a boil over medium-high heat. Cook until the barley is tender, stirring once or twice, 25 to 35 minutes. Drain the barley and cool completely. The barley can be prepared up to 3 days in advance; cover and refrigerate before using.
Step 2In a large bowl, toss the greens with 1/4 cup dressing. Divide the greens among 4 serving bowls. Mix the barley, green onions and 1/4 cup dressing in the same bowl. Spoon the barley atop or beside the greens, dividing evenly. Place the radish slices in a small bowl and sprinkle with salt. Add the vinegar and sugar and toss until the radish slices soften slightly, about 30 seconds. Divide the slices among the bowls. Top with avocado slices. Drizzle the remaining dressing over the bowl. Sprinkle with cashews and sesame seeds and serve.