Make the dough: Combine all the ingredients except the flour for dusting in the bowl of an electric mixer outfitted with a dough hook and mix on medium speed until a smooth, elastic dough forms, about 8 minutes. Set aside in an oiled bowl covered with plastic wrap for one hour.

Make the filling: Heat the olive oil in a wide, deep skillet over medium heat and sauté the onions and scallions until wilted and glistening, about 10 minutes. Stir in the pan occasionally. Remove the onions and add the zucchini to the skillet, with a little extra olive oil if needed. Sauté until the zucchini no longer exudes water. Place in a mixing bowl with the onions.

Add the herbs, feta, and trahana, rice, or bulgur. Season to taste with salt and pepper. Set aside to cool while you roll out the dough.

Preheat the oven to 350F/180C. Lightly oil a 10-inch / 22.5-cm-round baking pan.

Lightly dust a work surface with flour. Using a dowel or rolling pin, roll out the pastry to a circle about 14 inches / 32 cm in diameter. Place the dough circle inside the oiled pan, edges hanging over the rim. Brush generously with olive oil. Place the filling inside and pat down. Fold in the excess dough so that it covers most of the filling, leaving an open area of exposed filling in the center that’s about 4 inches / 10 cm in diameter. Brush the surface of dough with olive oil.

Bake the pie in the lower third of the oven. About half way through baking, brush with the egg-white wash. The pie needs about 50 minutes total baking time.

Remove and cool slightly before serving.