Preheat the oven to 350°F.
Wash the eggplants and cut them lengthwise in half. Scoop out the flesh with a spoon, leaving about a ½-inch border all around. Roughly chop the reserved flesh and set it aside.
Sprinkle the eggplant shells with salt and place them cut-side down on paper towels for 15–20 minutes to draw out excess moisture. Pat them dry with more paper towels.
Line a baking sheet with parchment paper. Arrange the eggplant shells cut-side up, brush each with 1 tablespoon extra-virgin olive oil, and bake for 20–25 minutes, until tender.
Meanwhile, heat the remaining 1 tablespoon olive oil in a large skillet over medium heat. Add the chopped onion and cook for 5–7 minutes, stirring occasionally, until softened and translucent. Add the minced garlic and cook for 1 more minute, until fragrant.
Add the ground beef to the skillet, breaking it up with a spoon, and cook for 8–10 minutes, until browned. Stir in the chopped eggplant flesh and cook for another 5 minutes.
Pour in the crushed tomatoes, reduce the heat to low, and simmer for 10–15 minutes, stirring occasionally, until the sauce thickens. Season with salt and freshly ground black pepper to taste.
While the sauce is simmering, make the béchamel: In a medium saucepan over medium heat, melt the unsalted butter. Sprinkle in the all-purpose flour and whisk continuously for 1–2 minutes to form a roux.
Gradually whisk in the warm whole milk and continue cooking for 5–7 minutes, until the sauce is smooth and has thickened. Season with salt, freshly ground black pepper, and a pinch of ground nutmeg.
Turn the oven temperature up to 400°F. Remove the eggplant shells from the oven and spoon the meat mixture evenly into each shell.
Top each stuffed eggplant with the béchamel sauce and sprinkle with shredded cheese.
Return to the oven and bake for 15–20 minutes, until the cheese is golden and bubbly. For extra browning, broil on high for 2–5 minutes, watching carefully.
Serve hot. Store any leftovers in the refrigerator for up to 3 days and reheat in the oven or microwave.