Byrek (Albanian Cheese Triangles)

I don't know how to pronounce the name of these triangles of cheesy, flakey goodness, but that doesn't matter. They're delicious, and versatile, and shockingly simple to put together, and the recipe makes roughly 12 triangles; would you believe me if I told you I ate 10 of them?
Regardless, this recipe comes from the loveliest baking/cooking/food magazine I have ever laid my eyes upon: Sift, a publication courtesy of none other than the great King Arthur Flour. If you have a chance (which I know you do), order the fall issue here, and prepare yourself for a longing to bake like none other before, to spoon and sweep flour all day long, to roll pie dough and visit a grist mill and take a shot at steaming bread and -- and, to make byrek.

Isn't it beautiful?! I know. I thought it looked even better on a pastry board with some flour all over it.
Without further ado, may I present to you:
Byrek
Yields about 12 triangles
For the Dough:
1 1/2 cups unbleached all-purpose flour (spoon and sweep, my friends! Spoon and sweep!)
Pinch of salt
1/4 cup plus 2 tablespoons warm water
2 tablespoons olive oil
For the filling:
1 large egg
1 1/2 cups crumbled feta cheese
1/4 cup whole milk
2 tablespoons unbleached all-purpose flour
1 tablespoon olive oil
Final Touches:
1 large egg
Using a wooden spoon, mix the flour, salt, water and 2 tablespoons of olive oil in a large bowl, and stir until a dough forms. Transfer to a lightly floured surface (parchment paper preferred) and knead for about 4 minutes, or until the dough is soft and elastic, and doesn't stick to your fingers. Place dough back into the large bowl, cover with a towel, and let sit for 10 minutes while you make the filling.
Preheat oven to 375 degrees F. Briefly whisk the egg in a small bowl, and then add the feta, milk, flour, and olive oil. Whisk away. Line a baking sheet with parchment paper.
Take the dough from the bowl and place it on your floured surface; or, if you prefer (like I did), use a bit of olive oil on your hands and your surface to prevent sticking instead of flour. This will ensure a truly divine little flakey square, and you don't have to worry about drying out the dough. Divide dough into 3 equal parts, using one at a time, while the others sit pretty in plastic wrap waiting their turn. Use a floured rolling pin to roll dough into a very thin 12-inch square.
Brush the square with 1 tablespoon of olive oil, and cut dough into 4 equal strips, each 3 inches wide or so. This proved to be a bit of a challenge for me: since the dough is so elastic, it will pull a bit when cut. Do your best to make a smooth cut in the dough, and then carefully and slowly tear down the line the knife left.
Put 2 tablespoons of the filling at the upper left corner of the strip of dough, and fold as you would a flag: starting from the left corner, make a diagonal triangle to the right edge, and then down, and then to the left edge, down, etc. Do this until there is none left to fold (it's okay if a bit of the filling spills out). Place the triangles seam-side down on the baking sheet. Lightly whisk the egg and use a pastry brush to generously coat each triangle. Bake for 30-35 minutes. Cool for at least 15 minutes before stuffing your face with 10 of them.

If we're being totally honest here, my byrek might have been just a bit too thick; however, it was still incredibly delicious, and even more delicious in an arugula salad with heirloom cherry tomatoes, sesame ginger vinaigrette, and a healthy dose of parmesan. How do you say "YUM" in Albanian?