Put the eggplant cubes in a colander, salt lightly, and let sit for 20 minutes to draw out bitterness. Pat dry with a towel.
Heat the olive oil in a 30 cm / 12-inch heavy-bottomed pot over medium-low heat. Add the onion, celery, and garlic, and cook 6 to 8 minutes until soft and translucent.
Add the potatoes, stirring to coat them in the oil. Cook for 5 minutes, stirring occasionally.
Add the peppers and eggplant. Stir, cover, and cook on low heat for 10 minutes, stirring once or twice.
Add the tomatoes, salt, and pepper. Stir, cover, and simmer on low for 25 minutes, stirring occasionally, until the potatoes are almost tender when pierced with a knife.
Add the zucchini, cover, and cook 12 to 15 minutes more until all the vegetables are tender but still hold their shape.
Remove the lid for the last 5 minutes if the stew looks watery, to let the juices reduce into a light sauce.
Take the pot off the heat and stir in the torn basil. Rest for 10 minutes, taste and adjust salt, then serve warm or at room temperature.