For dressing, in a food processor bowl or blender container, place cut-up strawberries, olive oil, vinegar, ginger, salt, and pepper. Cover and process or blend until smooth. Set aside.
Prepare charcoal grill or preheat gas grill for medium heat. Place chicken on lightly greased rack of uncovered grill. Grill directly over medium heat for 12 to 15 minutes or until tender and no longer pink, turning once halfway through grilling. Let chicken stand for 10 minutes or until cool enough to handle. Cut into irregular pieces about 2x1 inches.
Meanwhile, combine spinach, sorrel or arugula, onion, and basil in a large salad bowl. To serve, divide spinach mixture among four dinner-size shallow bowls or dinner plates. Arrange fruits and chicken on greens. Drizzle strawberry dressing on each serving. Makes 4 main-dish servings.