Heat oven to 400°F. Grease with shortening or spray large cookie sheet with cooking spray. Unroll dough; place on cookie sheet. Starting at center, press out dough into 14x12-inch rectangle to form crust. Bake 10 minutes.
Meanwhile, stir onion into barbeque sauce with shredded chicken. Microwave on Medium (50%) 4 minutes, stirring once halfway through heating.
Remove partially baked crust from oven. Spoon hot chicken mixture over crust to within 1/4 inch of edge. Sprinkle feta cheese and Swiss cheese evenly over top.
Return to oven; bake 6 to 11 minutes longer or until crust is deep golden brown and cheese is melted and bubbly. Cut into squares to serve.