Preheat the oven to 425°F. Line 2 large baking sheets with parchment paper. 

Toss the eggplants in a large bowl with ¾ teaspoon of salt and a good grind of pepper. 

Put the flour into a shallow dish. Put the eggs and yolks into a separate dish with the 2 tablespoons of milk or water and whisk together well. Put the breadcrumbs into a third dish with 1 ¼ teaspoons of salt, mixing to combine. 

Working with one slice at a time, coat the eggplant in the flour, shaking off the excess, followed by the egg wash, then the breadcrumbs. Transfer to the lined sheets and continue with the rest. Drizzle the slices all over with 2 tablespoons of the oil per sheet and bake for 15 minutes. Switch the sheets around, then bake for 15 minutes more, or until nicely golden. Remove from the oven and turn the heat down to 400°F. 

Meanwhile, make the tomato sauce. Put the oil into a large saucepan over medium-high heat. Once hot, add the onion and cook, stirring occasionally, for 10 minutes until softened and lightly colored. Add the garlic and spices and cook, stirring continuously, for 1 minute more, then add the canned tomatoes, sugar, water, 1 ¾ teaspoons of salt, and a good grind of pepper. Bring to a simmer, then turn the heat down to medium and cook for 25 minutes, stirring occasionally, until thickened slightly. Stir in the cilantro, then measure out 3 cups of sauce. (You'll use this to build the pie.) Keep the rest in the pan, to warm up when serving. 

In a bowl, toss together the kataifi, half the pecorino, and the remaining 3 tablespoons of oil. 

Next, assemble the pie. Line a 9-inch springform cake pan with a piece of parchment paper large enough to cover the bottom and sides. Cover the bottom with a third of the eggplant slices (cutting them to fit, as needed). Top with a third of the sauce, a third of the mozzarella, and a third of the remaining pecorino. Continue in this way with the remaining eggplant, sauce, and cheeses. Lastly, top evenly with the kataifi mixture and bake for 35 minutes. Turn the heat up to 425°F and bake for 15 to 20 minutes more, or until nicely golden on top. Remove from the oven and let rest for about 30 minutes. 

Remove the outer ring of the pan and use the paper to help you lift the pie onto a board. Heat up the extra sauce and serve the pie warm, or at room temperature, with the extra sauce alongside.