Heat 2 tablespoons of the oil in a large frying pan over medium heat until shimmering. Add the onions, season with salt, and cook, stirring occasionally and adjusting the heat as needed, until completely softened and light golden-brown, about 30 minutes. Meanwhile, prepare the rest of the vegetables.

Trim the eggplant and zucchini, cut into 1-inch cubes, and place in the insert of a 6-quart or larger slow cooker. Trim, core, and cut the bell peppers into 1-inch dice and add to the slow cooker. Core the tomatoes, cut into 1 1/2-inch dice, and add to the slow cooker. Finely chop the garlic and add to the slow cooker.

When the onions are ready, add the tomato paste to the pan and stir to coat the onions. Transfer the onion mixture to the slow cooker. Add the remaining 2 tablespoons oil and 1/2 teaspoon salt. Stir to coat all the vegetables.

Cover the slow cooker and cook until the vegetables are extremely tender, 4 hours on high or 5 to 6 hours on low. If you'd like to cook off some of the excess liquid in the slow cooker, cook uncovered for the last 30 minutes.

Stir in the basil and taste for seasoning, adding more salt as needed. Drizzle with more olive oil before serving if desired.