Heat oven to 425°F. Spray large cookie sheet and 10-inch nonstick skillet with cooking spray. Add apple and onion to skillet; cook over medium heat, stirring occasionally, until apple is tender and onion is translucent. Stir in brown sugar. Cook and stir an additional 2 minutes. Remove from heat.
Unroll dough; place on sprayed cookie sheet. Starting at center, press out dough into 15x10-inch rectangle. Place provolone cheese slices in 5-inch-wide strip down center of dough. Spoon barbecue sauce with shredded chicken over cheese. Layer apple-onion mixture over chicken. Arrange Cheddar cheese blend slices evenly over apple-onion mixture.
With scissors or sharp knife, make cuts 1 inch apart on long sides of dough rectangle to within 1/2 inch of filling. Alternately cross strips diagonally over filling; turn ends under and press to seal. Brush beaten egg white over top.
Bake at 425°F. for 11 to 15 minutes or until crust is deep golden brown. Let stand 5 minutes before serving. Remove from cookie sheet; cut into crosswise slices.