Salt eggplant halves cut-side up for 15 minutes. Pat dry. Brush with olive oil. Roast face-down at 400F (200C) for 25-30 minutes until soft. Let cool slightly.
Make the kima: Brown ground meat in olive oil over medium-high. Add onion and garlic, cook 3-4 min. Add tomatoes, tomato paste, cinnamon, allspice, bay leaf. Simmer 20 min until thick and dry. Season.
Make the bechamel: Melt butter, whisk in flour, cook 1-2 min. Gradually add warm milk, whisking until thick. Off heat, stir in nutmeg, salt, white pepper, egg yolk if using, and cheese.
Scoop a little roasted eggplant flesh from each half to create a well. Mix scooped flesh into kima. Fill each eggplant shell with kima.
Top each filled eggplant with a generous spoonful of bechamel.
Bake at 375F (190C) for 20-25 minutes until bechamel is golden and puffed. Rest 10 minutes before serving.