Place an oven rack in the center of the oven. Preheat the oven to 450 degrees F. Spray 2 heavy baking sheets with vegetable oil cooking spray or line with silicon baking mats. Set aside.
In a medium bowl, mix together the flour, salt and pepper. Place the eggs in another medium bowl. Place the breadcrumbs in a third medium bowl. Toss the eggplant slices and mushrooms in the flour mixture to coat. Working in batches, dip the eggplant slices and mushrooms first in the eggs and then into the breadcrumbs to coat. Arrange in a single layer on the prepared baking sheet. Drizzle with the olive oil, if using, and roast until brown and crisp, about 20 minutes.
To assemble the sandwiches, spread each slice of bread with 2 tablespoons of tomato-basil sauce. Place the mushrooms on top. Add 1/4 cup of cheese and 1/2 cup of arugula. Place the eggplant slices on top. Spoon 1 tablespoon of tomato-basil sauce on top of the eggplant. Repeat for the remaining sandwiches and serve.