Activate the yeast. Stir the yeast and sugar into about one-third of the measured lukewarm water. Leave for 5–10 minutes, until bubbly. If it shows no activity, check the yeast before continuing.
Mix and knead. Add the yeast mixture, remaining water, dough oil, and salt to the flour. Mix until no dry flour remains, then knead for 8–10 minutes until elastic. Keep the dough soft and slightly tacky; do not work in extra flour just to make it dry.
Let the dough rise. Use a little shaping oil to coat the bowl, cover, and leave in a warm, draft-free place for about 60 minutes, until noticeably expanded and airy. A cool kitchen may need longer.
Divide and rest. Lightly oil the work surface and hands. Ease the dough out without punching it down. Make two equal pieces for every serving selected above, keeping each piece about 3.8 oz (107 g). Round loosely, cover, and rest for 15 minutes.
Heat the oil safely. During the short rest, heat the frying oil in the deep pot to 340–350°F (170–175°C), checking with a thermometer. Aim for about 1 inch of oil while keeping the pot no more than one-third full. Keep water away, use dry utensils, and stay beside the stove.
Stretch by hand. With lightly oiled hands, gently pull one piece from the center outward into an irregular flatbread about 5–5½ inches (12–14 cm) across. Keep the middle thinner than the rim without tearing it. Do not use a rolling pin. Shape the next pieces as space becomes available.
Fry in small batches. Lower the dough gently into the oil away from your body. Fry only one or two flatbreads at a time, about 60–90 seconds per side, until golden and puffed. Keep the oil at 340–350°F (170–175°C), allowing it to recover between batches. More dough requires more batches, not a deeper pot of oil.
Check and drain. Lift out and drain briefly on a rack or paper towels. Let the first flatbread cool enough to handle and open it: the center must have a set, airy crumb with no wet or gummy dough. If it browns before the center cooks, reduce excessive heat and stretch the next pieces more evenly. Do not taste raw dough.
Serve warm. Finish frying the remaining pieces in small batches and serve promptly, allowing them to cool briefly before eating. Offer cheese, kajmak, ajvar, jam, or honey separately; toppings are not included in nutrition.