In a small bowl, dissolve the yeast in the milk. Stir in the egg yolks.
In a large bowl, stir together the flour, salt, and powdered sugar. Mix in the butter and lard with your fingers until the mixture is crumbly. Add the milk mixture.
Knead the dough, but stop immediately when it comes together. Form two 270g balls and pat the balls into discs. Wrap each disc in plastic wrap and refrigerate for one hour.
Meanwhile, prepare the filling: In a large saucepan, bring the milk with the honey and granulated sugar to a boil. Stir in the ground poppy seeds, lemon peel, ground cloves, applesauce and cookie crumbs. As soon as it comes together, remove it from the heat and let cool to room temperature.
Line a baking sheet with parchment paper.
Turn one piece of the dough onto a lightly floured work surface and roll out to a 20x30-cm (~8x12-inch) rectangle. Use an offset spatula to spread half of the filling over the dough. Leave 1 cm (~1/3 inch) empty edge on every side of the rectangle. Fold over the edges. Starting with the longer side, roll the dough tightly into a long log. Repeat with the other dough.
Transfer the rolls to the prepared baking sheet. Brush with egg yolks and refrigerate until the surface dries (about 30 minutes).
Brush with egg white. Make a few small holes on top of the dough with a fork. Sprinkle the doughs with the whole poppy seeds and refrigerate until the surface dries (approximately 15 minutes). Meanwhile, preheat the oven to 200°C/390°F.
Bake the begjli for 15 minutes, then lower the oven temperature to 190°C/375°F. Bake for 10 more minutes, until the tops are browned and nice cracks show on the top. Remove from the oven and let cool to room temperature. (For longer storage, wrap the rolls tightly in plastic wrap and keep them in a dry, cool place for up to two weeks.)