From the sifted flour, mixed with the salt, egg, oil and water, knead a soft phyllo pastry dough. Place the dough onto a sprinkled with flour counter and divide it into 2 equal parts, shape them into balls.
Sprinkle them with flour and wrap them with a clean towel for about 20 min, in order for the dough to relax. Melt the butter (don't let it get hot though).
Take 1 ball and roll it out into a thin sheet using a rolling pin. Put an 8″ (20 cm) in diameter bowl in the middle of the sheet with the opening facing down in order to mark the circle and then cut out trapezoid-shaped pieces from the rest of it (see gallery). Remove the bowl and smear the marked circle with oil.
Start pulling the trapezoidal sheets toward the circle with your fingers 1 by 1, while also smearing them with butter in between. What you're going to get is a round, layered dough. Transfer it to a plate with about the same diameter and put it in the fridge for the butter to harden while you get to work on making the next ball (repeat the same steps as the 1st).
Take the 1st ball out of the fridge and put the 2nd one in its place.
On a slightly oiled counter, roll it out into a sheet (can use a rolling pin or your hands, while pulling and stretching the dough) about 12 1/2″ (32 cm) in diameter, the same size as the oven dish (pre-greased) you're going to be baking in. Place the sheet in the oven dish and sprinkle with the crumbled feta cheese.
Repeat this process with the other sheet and lay it on top of the feta. Cut the formed phyllo pastry into squares, then smear with the remaining butter on top.
Bake in a preheated 392°F (200 °C) oven for about 40 min. to a golden color.
Sprinkle the baked phyllo pastry with a little water and cover it with a clean towel to smother for about 10 min. Serve.