Cook sausage until browned, stirring often and crumbling with wooden spoon. Drain.
Place one can of dough rolls or one sheet of dough in bottom of a greased 9 x 13 inch baking dish. Pinch edges to seal if necessary.
Spread aioli over dough in an even layer with the back of a spoon. Sprinkle sausage over all.
Top sausage with an even layer of cheese. Place second dough sheet on top of cheese, pinching edges to seal if necessary. Brush beaten egg over dough top for a shiny baked finish if desired. Cover and refrigerate until baking if making ahead.
Bake at 375 degrees for 20-30 minutes until dough is cooked through and golden. Cut into squares to serve.
Notes
*substitute 1/2 cup mayonnaise and 1/2-1 cup cooked crumbled maple bacon for aioli