Preheat the oven to 325 degrees F (165 degrees C). Unroll pie crust and press into an 8-inch pie dish; set aside.
Heat olive oil in a skillet over medium heat; cook and stir green onions in hot oil until wilted and beginning to brown, 2 to 3 minutes. Remove from heat. Spread ham in pie crust and top with green onions.
Whisk egg substitute, evaporated milk, and mozzarella cheese in a bowl. Pour over ham mixture. Sprinkle with Parmesan cheese and chives.
Bake in the preheated oven for 20 minutes. Increase heat to 350 degrees F (175 degrees C) and continue baking until quiche is browned, 15 to 20 minutes. Remove from the oven and allow quiche to rest until eggs are set, about 30 minutes more.