The day before serving, slice the eggplants lengthwise into 1/4inch slices. Sprinkle with the salt and lay the slices on top of one another in 1 large colander or in 2 small colanders. Place a dish on top and weight it down (a large can of tomatoes works well.) Drain for 20 minutes. Dry the eggplant slices with paper towels.
Preheat the oven to 425 degrees F. Line a baking sheet with parchment paper if you're not using a non-stick pan. Brush the paper and eggplant slices with the olive oil and cook in the over for 25 minutes, until lightly brown. Set aside.
For The Bechamel Sauce:
Bring the milk to a soft boil in a large saucepan. In another large saucepan, make a roux by stirring butter and flour over medium heat until they form a thick paste. Add the nutmeg to the boiling milk and then pour it, a little at a time, into the warm roux, whisking vigorously to prevent lumps. When all the milk has been added, continue whisking over medium heat until the sauce thickens, about 3-5 minutes. Set aside.
For the Filling:
Bring the water to a boil and cook the pasta until just tender. Drain well, return to the pot and toss with the 2 tablespoons of olive oil. Set aside.
Grease a deep 9x17 inch baking dish and coat with the bread crumbs. Line the pan with the cooked eggplant slices, edge-to-edge, and hanging over the sides of the pan. (It's okay if there are spaces showing between the slices.) About 9 slices should be enough to cover the dish.
Mix the cooked pasta with 3 cups of the ragu sauce, 1 cup of the bechamel sauce and half of each the cheeses. Stir well. Pour half the pasta mixture into the prepared dish. Top with the ham, the remaining cheeses, ragu sauce, bechamel sauce and pasta mixture.
Fold the eggplant slices over the top and cover with approximately 5 more slices. Cover the dish with plastic wrap and place a baking sheet on top with 3 large cans to weight it down. Place in the refrigerator overnight.
The day of serving, preheat the oven to 400 degrees F. Remove the plastic wrap, re-cover the dish of pasta with aluminum foil and bake for 1 hours. Remove the foil and bake for another 30 minutes. Remove from the oven and let rest for 15 minutes.
To serve, invert onto a large serving dish a cut into wedges or thick slices.