Brush a large 18" x 26" baking sheet with a little bit of olive oil and then arrange the eggplant slices on it.
Brush the tops of the slices with more olive oil, sprinkle generously with salt and pepper and bake in the preheated oven for 15-20 minutes, flipping the slices once halfway through cooking, until the eggplant is softened and slightly golden. Remove and let cool until ready to use. Once eggplant is out of the oven, lower temperature to 350°F
While the eggplant is baking, heat some olive oil (or other cooking fat of your choice) in a large saute pan set over medium-high heat. Add the onion and garlic and cook until fragrant and slightly browned, about 3 to 4 minutes. Add the ground lamb and salt and continue cooking, stirring from time to time and breaking the meat as you go, until the meat is no longer pink, about 5 minutes.
Add the crushed tomatoes, water, dried oregano, black pepper, cinnamon and allspice; stir well, cover loosely lower heat and simmer for about 15 minutes.
While the sauce is simmering, place the cauliflower florets into a steam basket and cook over salted boiling water until really tender, about 8 to 10 minutes. Remove from heat and let cool slightly.
Carefully transfer the cooked cauliflower to a high-speed blender, add the milk, salt, pepper and nutmeg and process on high speed until smooth and creamy. While the motor is running, add the egg yolk trough the feeding hole in the lid and continue to mix until it's fully incorporated.
Remove the lid, add the grated cheeses and then resume processing until it's completely incorporated, about 15 seconds.
Spread about 1/3 of the meat mixture at the bottom of a 9" x 13" baking dish, then arrange half of the eggplant slices over it.
Top with the rest of the meat mixture, followed by remaining slices of eggplant, and finally, all of the bechamel sauce.
Bake in the 350°F oven for 40 to 45 minutes, or until the top gets nice and golden.
Let stand for at least 10 to 15 minutes before serving; garnish with a handful of chopped parsley, if desired.