Put the eggplant pieces in a large colander and sprinkle all over with the salt. Place the colander in the sink to drain for 30 minutes.
Preheat the oven to 425°F (220°C). Position a rack in the center of the oven.
Pat the eggplant dry with a clean kitchen towel and spread it in a single layer onto a rimmed baking sheet. Pour half of the olive oil over the eggplant, season with a bit of salt and pepper, toss, and again spread out the eggplant on the sheet pan. Roast, tossing 2 to 3 times during cooking, until the eggplant is deep golden brown and quite tender but still holding its shape, about 30 minutes. Remove to a large bowl; set aside.
Using the same baking sheet, add the red onion and red pepper, drizzle with half of the remaining olive oil, and season with salt and pepper. Spread the vegetables in a single layer and roast until tender and charred in spots, tossing a few times during cooking, 18 to 20 minutes. Add the onion and peppers to the bowl of eggplant.
In a large bowl, combine the remaining olive oil with the red wine vinegar and Dijon mustard. Whisk and season with salt and pepper. Add the arugula and gently toss to combine. Top with the roasted vegetables, avocado, and capers and finish by drizzling the lemon juice over everything. Serve.