Toss eggplant with salt in a colander. Place a small bowl underneath and let sit for 30 minutes, or refrigerate for 8 hours.
Rinse eggplant lightly under cold water and blot dry with paper towels.
Heat 2 tablespoons olive oil in a large, nonstick skillet over medium-high heat until it starts to shimmer. Add eggplant and stir-fry until cooked through and browned, 7 to 10 minutes. Transfer to a serving dish.
Combine shallots, parsley, basil, capers, lemon juice, remaining 2 tablespoons olive oil, balsamic vinegar, and garlic in a food processor. Pulse a few times to combine, then toss the mixture into the skillet. Sauté over medium heat for about 5 minutes, then stir in sun-dried tomatoes.
Pour sauce over eggplant and serve immediately.
