In a mixing bowl, combine warm water, warm milk, sugar, and active dry yeast. Let it sit for about 5 minutes until the yeast becomes frothy.
Add the vegetable oil and salt to the yeast mixture and give it a quick stir.
Gradually add the all-purpose flour to the wet ingredients, mixing well until a soft and slightly sticky dough forms.
Knead the dough on a floured surface for about 5-10 minutes until it becomes smooth and elastic. If the dough is too sticky, you can add a little more flour as needed.
Place the dough back into the mixing bowl, cover it with a clean kitchen towel or plastic wrap, and let it rise in a warm place for about 1 to 1.5 hours or until it doubles in size.
Preheat your oven to 180°C (350°F).
After the dough has risen, punch it down to release any excess air. Divide the dough into small equal-sized pieces, depending on the desired size of your poğaça.
Take one piece of dough and flatten it with your hands or a rolling pin. Add your desired filling (cheese, spinach, potatoes, etc.) in the center and fold the dough over the filling to enclose it.
Pinch the edges of the dough to seal the filling and shape the poğaça into a ball or any other desired shape (triangle, boat, half-moon, etc.).
Place the poğaça on a baking tray lined with parchment paper, leaving some space between each piece to allow for expansion.
Beat the egg in a small bowl and brush the top of each poğaça with the beaten egg. Sprinkle sesame seeds or nigella seeds on top for added flavor and appearance.
Bake the poğaça in the preheated oven for about 20-25 minutes or until they turn golden brown.
Once they are baked, remove the poğaça from the oven and let them cool slightly before serving.