Heat olive oil in a deep pan over medium heat. Add onions and cook until translucent, about 5 minutes. Add cinnamon stick, ginger, saffron, and salt and pepper to taste. Stir in ⅓ cup water, then place squabs breast side down in pan. Reduce heat to low and cook, covered, for 30 minutes, turning once.
Remove squabs from pan and set aside to cool. Simmer onion mixture, uncovered, until pan juices are reduced by half, about 10 minutes. When squabs are cool enough to handle, remove and discard skin and bones with your fingers, then finely shred meat. Place meat in a bowl, then stir in 1 tsp. cinnamon and 1 tsp. orange-flower water. Set squab mixture aside.
Whisk eggs into reduced pan juices, then mix in remaining 2 tsp. cinnamon, parsley, sugar, remaining 3 tsp. orange-flower water, and almonds. Cook until eggs have scrambled and absorbed all juices, so that consistency is somewhat dry, about 2 minutes. Remove from heat and set aside.
Place 1 sheet phyllo on a clean surface and brush with butter; place a second sheet over it and brush with butter. Place buttered sheets centered over an 8'' plate. Repeat phyllo process three times, laying each pair of sheets at an angle over previous sheets to form an irregular bottom crust. Center squab mixture on crust and top with almond mixture. Prepare remaining 2 phyllo sheets as above and place atop almond mixture. Fold overhanging bottom sheets over to completely enclose filling. Brush top with clarified butter to seal.
Heat 2 tbsp. clarified butter over medium heat in a nonstick frying pan large enough to hold bestila. Carefully slide bestila into pan and fry until sides turn golden brown, about 3 minutes. Cover bestila with a large plate, then carefully flip it over onto plate. Add remaining 1 tbsp. butter to pan, slide bestila back into pan on uncooked side, and fry about another 2 minutes. Slide onto a serving dish, sprinkle all over with confectioners' sugar or in a decorative pattern as shown in photograph, and serve.