Char each eggplant directly on the flame of a stove over medium to medium-low heat, rotating often until the eggplants are completely cooked and tender on the inside and charred on the outside. Set the eggplants aside and let them cool enough to handle.

Peel and remove the skins from the eggplants and slice the eggplants in half lengthwise. Remove as many seeds as you can with your hands. Place the cleaned eggplants into a large colander set over a mixing bowl. Let the eggplants drain for about 30 minutes to 1 hour. 

Make the filling: Place the drained eggplants into a mixing bowl and add the feta cheese, olive oil, eggs and salt. Mix well until combined. Set aside.

Preheat the oven to 375 degrees F.

Grease a 9x13 inch baking pan with olive oil. 

Make the egg wash: beat 1 egg with 1 teaspoon of water in a small bowl and set aside. 

Prepare the dough: Put the olive oil, water, salt, baking soda and sugar into a small pot over medium heat and stir. Cook until the mixture just comes to a simmer and transfer the pot off of the heat. Immediately add the flour into the mixture and stir with a wooden spoon until a soft dough comes together. 

Assemble and bake the pie: Transfer the dough onto a floured surface and roll it into a bowl. Evenly divide the dough into two pieces. Roll out one of the pieces of dough into a rectangle that is about 11 x 15 inches or big enough to fit into the baking pan upto the upper rim edge. Place the rectangular piece of dough onto the baking pan and gently lay it onto the bottom of the pan and up to the edges. This will be the crust. Roll out the second piece of dough to a 9 x 13 inch rectangle and set aside. Pour the eggplant and feta mixture evenly on top of the piece of dough in the baking pan. Use a spatula to evenly spread out the filling. Carefully place the 9 x 13 inch piece of rolled out dough on top of the filling to match the edges of the pan and cover the pie. Use a sharp knife to make about 3 slits in the dough. Brush the top of the pie with egg wash and bake in the oven for 40-55 minutes or until the crust is golden brown.

Cut the pie into square slices and serve hot.