Preheat oven to 425 degree F. Lightly coat a 15x10x1-inch baking pan with nonstick cooking spray; set aside. Roll the stack of phyllo sheets into a cylinder shape. With a sharp knife cut phyllo roll crosswise into 1/4- to 1/2-inch strips. Gently separate phyllo into strips and spread evenly in the prepared baking pan. Coat phyllo generously with additional nonstick cooking spray. Bake, uncovered, for 8 to 10 minutes or until phyllo strips are golden brown.
Meanwhile, in a 12-inch skillet cook the green onion in olive oil over medium-high heat for 1 minute or until just tender. Add chicken; cook and stir until heated through. Remove skillet from heat. Add spinach; cherry tomatoes, if using; tarragon; and pepper. Toss to combine.
Divide phyllo among 6 serving bowls. Spoon spinach mixture over phyllo. Drizzle with balsamic vinaigrette. Serve immediately. Makes 6 servings.