Pour 2 tablespoons of the oil onto a baking sheet and dip each strip of eggplant into the oil to coat. Season the strips with salt and pepper. Roast the eggplant on the bottom shelf of the oven until tender and deep brown on the underside, about 25 minutes. Using a metal spatula, scrape up the eggplant and transfer to a plate. In a small bowl, combine the remaining 3 tablespoons olive oil with the balsamic vinegar, 1/2 teaspoon salt and 1/4 teaspoon pepper. In a large bowl, toss the arugula with 3 tablespoons of the dressing.
Transfer the salad to a platter and arrange the roasted eggplant on top.
Drizzle the eggplant with the remaining 2 tablespoons of dressing and serve.
