How to Make It
Preheat the oven to 350°. In a large skillet, heat 2 tablespoons of the olive oil until shimmering. Add half of the eggplant in a single layer, season with salt and pepper and cook over moderate heat, undisturbed, until browned on the bottom, about 3 minutes. Stir, cover and cook until the eggplant is tender, about 5 minutes longer. Scrape the eggplant onto a large plate and repeat with 2 more tablespoons of the olive oil and the remaining eggplant.
Lightly oil a large rimmed nonstick baking sheet. Sprinkle 1/4 cup of the Parmesan cheese into a 6-inch round on each side of the baking sheet. Bake for about 3 minutes, until the Parmesan is melted and golden brown. Let the cheese rounds cool for about 30 seconds to firm up. With a spatula, carefully lift each round off the baking sheet and drape it over an inverted 4-inch-wide bowl. Let the cheese stand until cooled and crisp, about 2 minutes. Repeat with the remaining Parmesan to make 2 more cheese cups.
In a small bowl, whisk the remaining 1/4 cup of olive oil with the vinegar and season with salt and pepper. In a large bowl, toss the mesclun, scallions, tomatoes and cooked eggplant with the dressing. Carefully spoon the salad into the Parmesan cups and serve at once.