Heat the oven to 400°F. Line 2 baking sheets with foil and grease each with 1 tablespoon oil. Slice the eggplant as thin as you can manage and put the slices on the prepared pans in a single layer. Sprinkle with the paprika and generously season with salt and pepper.
Put the eggplant in the oven and roast until the slices release their liquid, the pan is dry again, and they start to shrivel, 20 to 30 minutes. Turn the heat to 325°F and cook until the eggplant is dry, barely pliable, and releases easily from the foil, 5 to 15 minutes, depending on their thickness. Transfer the eggplant slices to a rack to cool and keep the oven on.
Carefully remove the foil from the baking sheet and add the bread. Drizzle with the remaining 4 tablespoons olive oil. Cook, turning once or twice, until the bread is golden and toasted, 5 to 10 minutes.
To assemble the sandwiches, spread the bread with mayonnaise if you like. Sprinkle the tomatoes with salt and pepper, then layer them with the eggplant; top with arugula and the other piece of bread, and serve right away.