Prep the aubergine: Cut the aubergine into ½ inch pieces. Transfer to a colander set over a medium bowl and sprinkle with 2 tsp. salt. Let sit 30 minutes to allow the aubergine to release some brownish liquid. After 30 minutes, squeeze the aubergine to remove excess moisture.
Chop the red bell peppers, courgette, onions into 1 inch chunks. Place all ingredients into a bowl and set aside.
Halve the cherry tomatoes, coarsely chopped the garlic and transfer to a bowl. Add juice of half a lemon. Set aside.
Heat 2 Tbsp. extra-virgin olive oil in a large Dutch oven over high heat. Once sizzling hot, working in batches, add half of the aubergine. Cook stirring every 3-4 minutes to ensure all sides are nicely browned and golden. Using a slotted spoon, transfer to a plate. Repeat this step with the remaining aubergine.
Heat ¼ cup extra-virgin olive oil over medium-high. Add ¼ cup capers and ½ tsp. red pepper flakes (optional). Cook, stirring, until bubbling subsides, about 30–60 seconds.
Over medium to high heat, add the red bell pepper, courgette, and onions, season with 1½ tsp. salt, and cook, stirring frequently, scraping bottom of pan, cook until vegetables are softened, 12–15 minutes.
Meanwhile, bring a large pot of generously salted water to a boil. Add the pasta to the boiling water when ready to add the aubergine to the pot. Boil to the pasta until al dense. Reserve 1 cup of pasta water. Then drain pasta in a colander.
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, about 10 minutes.
Return aubergine to pot and cook until vegetables are tender about 5 more minutes. Taste and season with salt.
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 wooden spoon. Add more pasta cooking liquid as necessary if pasta starts to stiffen. Remove from heat. Add juice from the other half of the lemon over. Add 1½ cups basil. Taste for salt and pepper.