Preheat the oven to 200°C (about 400°F).
Activate the yeast: take a little of the 1 cup warm water and stir together the 1 teaspoon active dry yeast and the 1 teaspoon sugar listed with the yeast. Let sit 5–10 minutes, until bubbly. If it does not become bubbly, the yeast should be replaced.
Prepare the filling: in a medium bowl crumble and combine the 1 ½ cups feta cheese with the ¼ cup chopped fresh parsley (or substitute nigella/black cumin if you prefer). Set aside.
Make the dough: in a large bowl combine the 4 cups all-purpose flour and the pinch of salt. Add the 1 egg, 3 tablespoons olive oil, the remaining warm water from the 1 cup (after reserving a little for the yeast), and the activated yeast mixture. Stir until a rough dough forms.
Knead the dough on a lightly floured surface (or in the bowl) until smooth and elastic, about 5–10 minutes by hand or until it comes together well.
First rise: place the dough in a lightly oiled bowl, cover with a clean towel or plastic wrap, and leave in a warm place for about 30 minutes, until it has roughly doubled in size.
Divide and rest: punch down the dough lightly, cut it into egg-sized balls, and cover them with a towel or plastic wrap. Let them rest 10 minutes to relax the gluten.
Shape and fill: working with one ball at a time, roll each ball into an oval about 1/8–1/4 inch thick. Place a spoonful of the cheese mixture in the centre of the oval (enough to fill but not overfill). Fold the dough over the filling and press the edges firmly to seal, forming the traditional fatayer shape.
Prepare for baking: arrange the filled pastries on a baking sheet lined with parchment or lightly greased. Brush the tops with a little olive oil (do not use an egg wash unless you add an extra egg to the ingredient list). If using, sprinkle sesame seeds on top.
Bake: place the baking sheet in the preheated oven and bake until the tops are golden, about 12–18 minutes (oven times vary).
Cool briefly and serve: remove from the oven, let cool a few minutes on a rack, then serve warm.