Step 1Cut the green tops off the onions, then cut the greens lengthwise into thin strips. Add these to a bowl of ice water. Save the white parts of the onion for another use.
Step 2Toss together the button and shiitake mushrooms on a large platter. Combine 2 tablespoons of soy sauce, 1 tablespoon of water, the Sherry, 2 teaspoons of sesame oil, 4 teaspoons of sugar and the garlic. Pour this over the mushrooms, tossing until the mushrooms are evenly coated. Let stand 10 minutes.
Step 3Meanwhile, heat an outdoor grill or stovetop grill pan. Grill the mushrooms until slightly browned around the edges, 5 to 8 minutes, turning as they grill.
Step 4While the mushrooms are grilling, make the dressing by combining the remaining 2 teaspoons of soy sauce, 1 tablespoon of water, the rice vinegar, the remaining 1/2 teaspoon of sugar, the remaining 1/2 teaspoon of sesame oil and the red pepper flakes. Drain the green onions and toss together with the mixed greens. Pour the dressing over the salad to coat. Serve the greens along with the grilled mushrooms.