Cut 1 large or 2 medium Italian eggplants into ½" pieces. Transfer to a colander set over a medium bowl and sprinkle with 2 tsp. salt. Let sit 30 minutes. Your eggplant will start to discolor and should start to release some brownish liquid that will fall into the bowl below. That’s good!
Meanwhile, cut 1 large or 2 small red bell peppers and 1 large zucchini into ½" pieces. Transfer to a medium bowl. Peel 2 medium onions, cut through root end, then slice through root end into thin slivers. Transfer to bowl with bell peppers.
Halve 1 pint cherry tomatoes (the fastest—but dorkiest—way to do this is to sandwich them between two quart container lids) and transfer to a small bowl.
Smash 4 garlic cloves, then coarsely chop and add to bowl with tomatoes.
Slice 1 lemon in half; set aside.
Finely grate 1½ oz. Parmesan with a Microplane or the small holes of a box grater into another small bowl.
Your eggplant needs special treatment: Cooking it separately guarantees that you can make it tender and golden (rather than chewy and gray) without over-cooking it or causing the other veg to get soggy. When eggplant has been sitting 30 minutes (okay, 25 if you’re antsy), gently squeeze it to get rid of any more moisture (less moisture means more browning). I like to layer it between kitchen towels and gently press it to remove the moisture.
Heat 2 Tbsp. extra-virgin olive oil in a large Dutch oven or heavy pot over high until shimmering. Once hot (test with one piece for starters!), add half of eggplant and stir to coat in oil. Cook, stirring every 3 minutes or so (you want to leave it undisturbed to get browning while also making sure lots of pieces get their time against the hot pan), until eggplant is golden, with some pieces starting to brown and char, 5–7 minutes. Reduce heat to medium if eggplant is browning too quickly. Using a slotted spoon or fish spatula, transfer to a plate. If any eggplant pieces stick, just scrape them up—no biggie. Heat 2 Tbsp. extra-virgin oil over high. Cook remaining eggplant using same process, 5–7 minutes. Transfer to plate with cooked eggplant.
Scrape off any bits of eggplant from pot if needed (no need to wipe out). Heat ¼ cup extra-virgin olive oil over medium-high. Add ¼ cup capers and ½ tsp. red pepper flakes (don’t be alarmed if capers sputter and foam—it’s because they’re wet!). Cook, stirring, until bubbling subsides, 30–60 seconds.
Add onion, season with 1½ tsp. salt, and cook, stirring frequently, scraping bottom of pan, and reducing heat if onions are getting too brown. Add peppers after onion has gone translucent and starting to brown. Continue cooking until vegetables are softened, slumped, and have started to take on lots of color, 12–15 minutes. Meanwhile, bring a large pot of generously salted water to a boil (this is for your pasta).
Add tomato-garlic mixture to sauce, reduce heat to medium-low, and cook, stirring occasionally with a wooden spoon, until almost all tomatoes have burst and vegetables are starting to break down and deeply caramelize, about 10 minutes. Keep an eye on it and reduce heat if veg is getting too dark for your liking.
Finish the sauce: Return eggplant to pot and cook until vegetables are tender but the vegetable pieces are still distinct (you don't want total mush), about 5 more minutes. Taste and season with salt.
Add pasta to boiling water right after you add the eggplant, stirring occasionally, until al dente. Before draining, scoop out 1 cup pasta cooking liquid with a liquid measuring cup.
Now the second more important part. Reduce heat over sauce to low. Add drained pasta and stir to coat in sauce. Add 1 cup Parmesan, pour ½ cup pasta cooking liquid over, then stir vigorously with a rubber spatula or wooden spoon until a creamy sauce forms with no clumps of cheese. Add more pasta cooking liquid as necessary if pasta starts to stiffen. Remove from heat. Squeeze juice from ½ lemon over, then stir in 1½ cups basil. Taste for salt and lemon juice.
Divide pasta among bowls. Top with cheese, basil, and red pepper flakes. Drizzle with oil. Don’t be sad if you can’t finish it all: It’s even better as cold leftovers.