Cut stripes in each eggplant's peel by running a vegetable peeler down the length of it and repeating at about 1-inch intervals.
Slice the eggplants into rounds 1/3 to 1/2 inch thick. Lay them on a baking sheet and sprinkle lightly with 1/2 teaspoon salt. Let stand for about 5 minutes.
Blot the eggplant slices with paper towels and lightly coat both sides with olive oil cooking spray. Grill the eggplant, flipping halfway through, until soft and browned on both sides, 9 to 11 minutes total.
Puree oil, vinegar, tomato, garlic, paprika and the remaining 1/4 teaspoon salt in a blender until well combined.
Toss salad greens with half the vinaigrette in a medium bowl.
Arrange the eggplant slices on 6 salad plates.
Drizzle with the remaining vinaigrette.
Place the salad greens over and between the eggplant slices, then scatter the cheese curls on top of each salad.