Process the basil, olive oil, pine nuts, garlic, and Parmesan in a food processor until finely ground, about 1 minute, scraping down the sides as needed.

Toss the eggplant with salt in a bowl. Line a large plate with a double layer of coffee filters and lightly spray with vegetable oil spray, then spread the eggplant evenly over the filters and microwave until the eggplant is dry to the touch and slightly shriveled, 8 to 12 minutes, tossing halfway through.

Heat oil in a Dutch oven over high heat until shimmering. Add the prepared eggplant, onion, and potatoes and cook, stirring frequently, until the eggplant begins to brown, about 2 minutes.

Push the vegetables to the sides of the pot. Add oil and tomato paste to the center and cook, stirring often, until a brown fond develops on the bottom of the pot, about 2 minutes. Stir in water and the tomatoes with their juice, scraping up the browned bits, and bring to a boil. Reduce the heat to medium, cover, and gently simmer until the eggplant is completely broken down and the potatoes are tender, 20 to 25 minutes.

Heat the remaining oil in a large skillet over high heat until just smoking. Add the zucchini, bell peppers, and salt and cook, stirring occasionally, until the vegetables are browned and tender, 10 to 12 minutes. Push the vegetables to the sides of the skillet, add the pesto to the clearing, and cook until fragrant, about 1 minute. Stir the pesto into the vegetables and transfer to a bowl; off the heat, add the remaining water to the skillet to scrape up any remaining browned bits.

Remove the Dutch oven from the heat and stir in the sautéed vegetable mixture and the flavored water from the skillet. Cover and let the stew stand for 20 minutes to allow the flavors to blend. Stir in the fresh basil, season with salt to taste, and serve hot.