Heat the oven to 350°F. Cut off the stem and bottom of the eggplants, and then cut them in half widthwise. Cut each half in half again from top to bottom so you have quarters (they will be almost square-shaped). Using a paring knife, cut as much flesh out of the skins as possible while still leaving the skins intact. (Make sure you don’t cut the skin and make sure to leave a tiny bit of eggplant still on, about 1/8 inch.)
Place the eggplant skins on a baking sheet. Sprinkle the skins with 2 tsp. salt and set aside for 20 minutes (the salt helps get out the bitterness and excess moisture).
Dice the eggplant flesh into 1/4-inch cubes, and place in a colander or sieve set over a bowl. Sprinkle with 1 Tbs. salt, mix together, and set aside for 20 minutes.
Break up the French bread into 2- to 3-inch chunks, place in a large bowl, and cover with cold water. Let soak for a few minutes to soften. Remove and discard as much crust as possible. Grab a handful of bread, squeeze out as much water as you can, and place it in another large bowl. Repeat with the remaining bread. Break up the soaked bread into smaller pieces (no bigger than 1/2 inch) and set aside.
Place a large (12- to 14-inch) sauté pan over high heat for several minutes. When hot, add the oil and heat until shimmering. Add the eggplant skins and sauté over high heat, turning now and then until just limp and pliable, about 3 minutes (you may have to do this in batches). Transfer to a paper towel–lined plate to drain.
Grab handfuls of the diced eggplant and squeeze out as much moisture as possible (and therefore any bitterness and extra salt), and transfer to another bowl.
Drain half the oil out of the sauté pan, reserving the drained oil. Place the pan over medium-high heat. When the remaining oil is hot, add half the squeezed eggplant and sauté until golden, about 5 minutes, stirring now and then. Add half the garlic and cook for another 2 to 3 minutes. Remove to a baking sheet to cool, and repeat with the other half of the eggplant and garlic, adding 3 Tbs. of the oil used to cook the skins before adding more eggplant.
Once cooled, add the sautéed eggplant to the bread pieces in the mixing bowl.
Add the beaten eggs, Parmesan, parsley, and 1 cup of the mozzarella to the bowl with the eggplant and bread. Mix well. Season with the remaining 1 tsp. salt and the pepper.
Cover the bottom of a 9x13x2-inch baking dish with 1/2 cup of pomodoro sauce. Cup one of the eggplant skins in one hand and fill it with about 1/3 cup of the stuffing. (The amount depends on the size of the eggplant, but it should be enough filling so they’re three-quarters full and look like a stuffed pasta shell. Try to pack in the filling so it doesn’t come out.) Repeat with the remaining skins, placing them side by side in the pan after they are filled (they should prop each other up but shouldn’t be crammed together).
Pour the remaining pomodoro sauce over the stuffed eggplants, cover the pan with foil, and bake for 30 minutes. Remove the foil, sprinkle the eggplants with the rest of the mozzarella, and continue to bake until the sauce is bubbly, the mozzarella is lightly browned, and the interior of the eggplant is hot, 15 to 20 minutes.