Combine the flour, water, salt, dry yeast, and sugar and knead for about 5 minutes until a smooth, elastic dough forms. Divide the dough into 8 equal portions and roll them out as thinly as possible. Combine the olive oil and melted butter and brush some of it over a large plate. Brush the mixture over both sides of each dough piece and then stack the dough pieces on top of each other on the greased plate. Let the dough rest for 30 minutes.
Meanwhile, prepare the spinach, pomegranate cream cheese, chicken, and cheese fillings. After 30 minutes, spread each dough piece out by hand as thinly as possible, spoon half of a filling along the edge of each dough piece, tightly roll up each dough strand, and place them all next to each other. In the end, you should have 2 spinach-filled dough strands, 2 pomegranate cream cheese-filled dough strands, 2 chicken-filled dough strands, and 2 cheese-filled dough strands.
Grease a metal rolling pin with butter, place it in the center of the stuffed dough strands, and drape the stuffed dough strands over the rolling pin. Braid the outer strands of dough so that you have 3 braided dough strands on either side of the rolling pin. Brush the egg wash over the dough and sprinkle sesame seeds on top. Transfer the stuffed börek braid to a preheated oven and cook at 375°F for 1 hour.
For the baba ganoush, cut the eggplants in half lengthwise, press the garlic slices into the eggplants, drizzle olive oil on top, and cook with the stuffed börek braid for 20 minutes. Once the cooked eggplants have cooled, combine the pulp with the remaining ingredients to form a cream. Once the stuffed börek braid has finished cooking, remove the metal rolling pin and pipe the baba ganoush into the opening.