- Place the diced eggplant in a colander and season with 1 teaspoon kosher salt. Toss well to coat, and allow the eggplant to sit for 15 minutes.
- Using your hands, squeeze the eggplant to remove any excess liquid. Place the eggplant on a clean dish towel. Wrap the eggplant in the towel and wring to remove any residual liquid. Set aside.
- Meanwhile, in a large bowl, whisk together the eggs, heavy cream, Parmesan cheese, and 1 teaspoon salt.
- Add the bread pieces to the bowl and, using a rubber spatula, submerge the bread in the custard. Allow the bread to soak while you prepare the vegetables.
- Oil a 9 x 13-inch baking dish with 1 teaspoon olive oil. Preheat the oven to 350 degrees F.
- Heat a large skillet over medium-high heat. Add the oil to the pan and swirl to coat the bottom.
- Add the dried eggplant to the pan and cook, stirring often, until the eggplant is golden brown and beginning to soften, about 5 minutes.
- Add the zucchini, red pepper, onion, garlic, red pepper flakes, and ¾ teaspoon kosher salt to the pan and stir to combine.
- Cook, stirring often, until the vegetables are fragrant, beginning to soften, and have just a whisper of golden brown, about 7 minutes.
- Stir in the sun-dried tomatoes and remove the pan from the heat. Allow the mixture to cool for about 5 minutes.
- When the mixture has cooled slightly, add the vegetables to the bowl with the bread, along with the torn basil, and stir to combine.
- Pour half of the mixture into the prepared pan. Sprinkle with half of the shredded mozzarella cheese.
- Top with the remaining bread mixture and sprinkle the remaining mozzarella on top.
- Bake the bread pudding for about 50 minutes or until the casserole is golden brown on top, bubbly around the edges, and a toothpick inserted in the center comes out clean.
- Allow the bread pudding to rest for 10 minutes before serving.