Heat oven to 400°F. Grease large cookie sheet with shortening or cooking spray. Unroll dough onto cookie sheet. Starting at center, press out dough into 15x10-inch rectangle. Brush oil over dough. Sprinkle evenly with Italian seasoning.
Bake 10 to 13 minutes or until edges are golden brown. Meanwhile, in medium bowl, mix chicken, onions and mayonnaise.
Cut baked crust in half crosswise. Spread chicken mixture on 1 half. Top with tomato and cheese, cutting to fit if necessary. Cover with remaining half of baked crust, seasoning side down. Cut into 4 sandwiches.
Heat 12-inch skillet over medium-low heat. Place 2 sandwiches in skillet; cook about 2 minutes, pressing down with pancake turner, until bottom crust is deep golden brown. Turn sandwiches; cook about 2 minutes, pressing again with turner, until cheese is melted. Repeat with remaining 2 sandwiches.