Preheat the oven to 350 degrees F and lightly grease a large baking dish or roasting pan.
Wash the eggplants well and cut in half lengthwise. (If eggplants are large and seedy, use a spoon or paring knife to remove the seedy portion, making sure to leave at least a 1/2-inch shell. Discard seedy flesh.) Using a fork, prick the inside well all over the cut sides. Season the eggplant with salt and pepper.
Place enough olive oil to come up 1/2-inch up the sides of a large saute pan and heat until hot. Add the eggplant halves, working in batches if necessary, and cook, cut side down, until tender and golden brown, 4 to 6 minutes. Carefully turn the eggplant halves over and continue cooking on the other side until the eggplants are very tender. Using a slotted spoon or spatula, transfer the eggplants to paper towels to drain. Remove as much of the oil as possible, then place the eggplant halves, cut side up, in the prepared baking dish. Repeat if necessary with any remaining eggplant halves, adding more olive oil as necessary.
While the eggplants are cooking, heat 2 tablespoons of olive oil in a large saucepan or Dutch oven and saute the beef and onions until golden brown, 6 to 8 minutes. Add 3/4 teaspoon of the salt, 1/4 teaspoon of the pepper and the white wine and cook until mostly reduced, 1 to 2 minutes. Add the tomato sauce, stir to combine, and cook until sauce is thickened and flavorful, 5 to 10 minutes. Remove from the heat, stir in 1 tablespoon of the parsley, taste, and adjust seasoning if necessary.
Make the bechamel: melt the butter in a small saucepan and add the flour. Stir to combine and cook until fragrant, 1 to 2 minutes; do not allow the flour to color. Working quickly, whisk in the milk until completely incorporated and smooth and cook until thickened, 4 to 6 minutes. Add 1/2 teaspoon of salt, a pinch of pepper, and 1/4 cup of the cheese. Stir, adjust seasoning if necessary, and remove sauce from the heat. Allow to stand for 1 to 2 minutes. In a small bowl whisk the eggs until smooth. Add the eggs to the hot bechamel and whisk quickly to combine. Use sauce immediately or set aside, covered, so as not to form a skin on top.
When all components are prepared, turn the eggplant halves cut side up in the baking dish. If desired, hollow out some of the eggplant flesh and add to the meat sauce. Divide the meat filling evenly among the eggplant halves. Divide the bechamel topping evenly among the eggplant halves and sprinkle with the remaining cheese. Bake, uncovered, for 40 to 45 minutes or until heated through and golden brown on top. Top with more bechamel and cheese, if desired. Sprinkle with some of the remaining parsley and serve immediately.