Prepare eggplant: Slice into 1/2-in.-thick rounds. Sprinkle both sides with salt and let sit 1/2 hour. Pat dry, brush with oil, and cook over medium heat in a large cast-iron skillet on both sides until cooked through, about 10 minutes total.
Meanwhile, make tapenade: Put garlic and anchovies to taste in a food processor and pulse until well chopped. Add olives, tomatoes, capers, basil, and pepper to taste and pulse to combine. Add oil and pulse until combined but still a little bit chunky.
Make sandwiches: Toss lettuce with a minimal amount of dressing. Spread 1 side of 2 bread slices with 2 tbsp. tapenade each. Brush other 2 slices with oil on 1 side, then turn over and top with most of brie, lettuce, red pepper strips, and eggplant, dividing evenly. Add remaining brie and top with tapenade-spread bread slices, with tapenade side down. Brush tops of sandwiches with olive oil and season with pepper.
Reheat cast-iron skillet over medium heat. Put sandwiches in skillet and cook, turning once until golden brown and cheese is gooey, about 5 minutes total.