Dissolve the sugar and a handful of flour + the yeast in the warm water to make a sludge, cover with plastic wrap and leave it in a warm area to rise.
Sift the flour into a large bowl, mix with the salt and stir. Set aside 1 cup flour for kneading, make a well in the rest and add the beaten eggs and risen yeast mixture. Knead a soft and stretchy dough, while gradually adding the set aside flour.
Roll out the dough onto a floured counter into a rectangular sheet 1/2″ (1 cm) thick. Smear it with the butter/fats, melted beforehand, and sprinkle with 1/3 of the crumbled feta.
Fold the sheet into a package, bringing its 4 corners together in the middle and flip it over onto the floured counter.
Roll out the sheet to its initial dimensions, smear generously with butter, sprinkle with another 1/3 feta and fold into a package again. Flip it over, roll out to its initial dimensions, smear with butter and sprinkle with the remaining feta.
Fold the sheet into a package and without rolling it out, flip it over onto a tray, covered with baking paper.
Cut the resulting square package as follows: first make a cross through the middle of the square, then cut across the diagonals of the square and then across the diagonals of the 4 resulting smaller squares (see photo).
Smear the tutmanik with the remaining butter and beaten egg.
Put the tray in a cold oven, turn on the bottom element at 338°F (170 °C) initially and once it gains a light golden color on the bottom, turn off the bottom element and turn on the top element to the same degrees, bake to a golden brown color on top.
Characteristic of this tutmanik is that rises as it bakes, the result is fantastic.