Preheat the oven broiler. Brush olive oil on the baking sheet and arrange the eggplant slices on it. Brush more oil on each slice. Place the baking sheet about 4 to 6 inches from the broiler and cook until lightly golden, turning once, about 5 to 8 minutes. Turn off the broiler and preheat the oven to 425° F/220° C.
Pat the pizza dough out until it’s about ¼-inch thick, and transfer to a pizza pan. Spread pesto all over the pizza, then arrange the eggplant slices evenly on the pesto. Dollop a little more pesto around the pizza, and then add the sun-dried tomatoes. Sprinkle on the Parmigiano cheese.
Bake in the preheated oven until the crust turns golden brown and the tomatoes and eggplant crisp a little around the edges, about 10 to 12 minutes. Remove from the oven and allow to cool about 2 to 3 minutes, then serve.