Prepare pizza dough according to package directions. Pat into a 15x10-inch rectangle on a baking sheet.
Heat oil in a large skillet over medium heat. Add onion and mushrooms and cook until tender.
Stir in eggs and chilies and cook until set. Remove from heat.
Sprinkle ½ cup cheese lengthwise down center of the dough, leaving about a 2½-inch border on each side. Spoon egg mixture evenly over cheese. Sprinkle remaining cheddar over egg mixture.
Fold sides of dough over filling. Brush with egg white if desired.
Bake for 15 minutes or until golden brown.
Let stand 5 minutes. Cut crosswise into slices.
Notes
Leftover onions can be used for pizza toppings or to add to scrambled eggs or chicken salad.