Preheat the oven to 400 degrees. Line a large baking sheet with a layer of paper towels and arrange eggplant in one layer over the paper towels. Sprinkle the tops of the eggplant with kosher salt and allow to sit and sweat out for 30 minutes. If you don’t have time, just skip this step, but it does take some of the bitterness out of the eggplant.
In the meantime, sauté mushrooms (if using) in a large skillet with 2 Tablespoons of oil. Cook until mushrooms have released their moisture and are golden brown, about 6-8 minutes. Season with salt and pepper to taste.
Make fresh tomato sauce (or use jarred): Bring a large pot of water to a boil. Add the tomatoes and turn the heat off. Blanch for 30 seconds or until skin starts to curl where you made the X. Remove with a slotted spoon and place on a cutting board. When cool enough to handle, core and peel skin off. Cut through the “equator” of the tomato (not through the core) and gently squeeze out the seeds. It’s ok if they don’t all come out. Chop the tomatoes coarsely.
In a large skillet, warm enough olive oil to cover the bottom with a thin layer, about 3-4 Tablespoons. Add the garlic and crushed red pepper and sauté until fragrant, about 30 seconds. Add the tomatoes and a big pinch of salt. Bring to a simmer and cook until a little thicker and the tomatoes are broken down, about 10-15 minutes. Taste for salt and blend with an immersion blender until still a bit chunky but more like a sauce. Stir in basil. This can be done several days in advance.
Blot the tops of the eggplant with a paper towel to remove the excess moisture. Remove eggplant from the baking sheet and line with unbleached parchment paper. You’ll need about 2 baking sheets. Brush both sides of the eggplant with ¼ cup of olive oil and arrange in one layer on the prepared baking sheet(s). Bake for 20-25 minutes, flipping halfway, until the eggplant is golden brown and tender. LOWER OVEN TEMPERATURE TO 350 DEGREES.
In a medium bowl, toss breadcrumbs with ¼ cup of olive oil. If the breadcrumbs have no flavor, add a pinch of salt. You can also add a handful of grated parmesan cheese if you want.
Spread a thin layer of the sauce on the bottom of a 13 x 9-inch baking dish or equivalent**. Arrange one layer of cooked eggplant over the sauce. It can overlap a little. Scatter a layer of mushrooms on top, if using. Spread another layer of tomato sauce over the vegetables. Repeat with eggplant and more mushrooms and sauce. Place the slices of mozzarella on top and then cover with the breadcrumb mixture. Bake for 15 minutes, until ingredients are warmed through and cheese is melted. If your breadcrumb topping is not golden brown, you can broil the top for a few seconds until golden.