This classic Turkish flatbread is usually cooked in a brick oven. To bring this homemade version as close as possible to restaurant style, we bake it at a high temperature on a baking sheet that’s been heating in the oven, providing the effect of cooking on the hot surface of the brick oven floor.
For the dough: Put 1 1/2 cups warm water (about 110 degrees F) in the bowl of a stand mixer with the paddle attachment. Add the flour, sugar, yeast and 1 teaspoon salt to the water and mix on low until barely incorporated. Let the mixture sit for 5 minutes for the flour to absorb the water. Attach the dough hook and knead the dough on medium high until smooth, about 7 minutes. Coat a large bowl with vegetable oil and transfer the dough to the bowl. Move the dough around so it’s coated with oil on all sides. Cover and let rise in a draft-free space, such as an unlit oven, until increased in size but not quite doubled, about 1 hour.
Uncover the dough and place on a well-floured surface.
Place a baking sheet upside down on the middle rack of the oven and preheat to 450 degrees F. Turn a second baking sheet upside down and cover with a piece of parchment.
Divide the dough into 5 pieces (see Cook's Note) and cover 4 with a damp cloth. Roll out the uncovered piece using a rolling pin into an oval about 20 inches long and 5 inches wide. Transfer the dough to the parchment on the back of the baking sheet. Repeat with a second piece of dough.
For the toppings: Sprinkle about 1 cup kaşar on each oval, leaving about a 1-inch border on all sides. Top the cheese with sucuk slices, leaving some space in the middle to add an egg later. Fold the dough up and over about 3/4 inch onto the cheese on all sides and pinch the ends. Brush with vegetable oil.
Slide the pides with the parchment onto the baking sheet in the oven. Bake until the cheese is melted and starting to brown, 7 to 10 minutes. Crack 2 eggs into 2 small bowls. Add 1 egg on top of each pide in the middle. Continue to bake until the eggs are set to your preference, another 4 to 5 minutes. Repeat with the remaining dough, cheese, sucuk, oil and eggs.
You can also divide the dough into 6 pieces and make slightly smaller pides.
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