In a large mixing bowl, whisk together the flour, sugar, salt and instant yeast. Add the olive oil and lukewarm milk, then mix with a spatula. Gradually pour in the water while stirring until a sticky dough forms.
Lightly oil a separate large bowl and transfer the dough into it. Cover with a clean kitchen towel and let it rise in a warm place until doubled in size, about 40–60 minutes.
Transfer the dough to a lightly floured surface. Fold the edges towards the centre without kneading, then shape it into a ball. Divide into 4 equal pieces (about 180 g / 6 oz each) and shape each into a ball. Cover with a kitchen towel and let them rest for 30 minutes.
Place an oven-safe bowl filled with water at the bottom of the oven and preheat to 250°C (480°F). Line a baking sheet with parchment paper.
Lightly flour your work surface. Take one dough ball and roll it out into a thin oval, about 28 cm / 11 inches long and 15 cm / 6 inches wide.
In a small bowl, whisk together the egg yolk and yogurt. Spread some of the mixture over the dough using your hand.
Press your fingertips firmly into the dough to create the traditional tırnak pide pattern. First, make indentations across the width, then along the length. Dip your fingertips into the egg yolk and yogurt mixture as needed. Make sure the indentations are deep enough so the pattern stays visible as the bread rises in the oven.
Dry your hands, carefully lift the shaped dough by both ends and transfer it to the prepared baking sheet.
Bake on the lower-middle rack for 8–10 minutes, or until golden brown.
Remove the pide from the oven and transfer it onto a clean kitchen towel. Cover loosely with another towel to keep it soft.
Repeat with the remaining dough balls, shaping and baking each pide in the same way. Serve warm.