Prepare the eggplant: Place the diced eggplant in a medium bowl. Sprinkle it with the salt. Using your hands, toss the diced eggplant until it is evenly covered with the salt.
Drain the eggplant: Place the salted eggplant in a colander over the sink. Allow it to drain for at least 15 minutes.
Cook the onion: In a medium nonstick skillet over medium heat, warm 2 tablespoons of water. Add the diced onion to the skillet and cook for about 6 to 8 minutes, or until softened. Add 1 tablespoon of water at any point to prevent sticking.
Cook the garlic: Add the garlic to the skillet and cook for 30 seconds to 1 minute, until fragrant.
Transfer the onion and garlic: Transfer the onion and garlic to the slow cooker.
Fill the slow cooker: Add to the slow cooker the drained eggplant, yellow squash, zucchini, red bell pepper, yellow bell pepper, diced tomatoes with their juices, tomato paste, cannellini beans, herbes de Provence, crushed red pepper, and freshly cracked black pepper, to taste. Stir well to combine. If your slow cooker is packed to the brim, it is easier to gently toss the ingredients with your hands instead of trying to stir with a spoon. (Don’t worry about filling your slow cooker close to capacity. As long as the lid is able to close, you’re good. The vegetables will get softer and cook down over time, taking up less space.)
Cook: Cover with the lid. Set the slow cooker to the Low heat setting for 5 hours.
Serve: After 5 hours, lift the lid. Stir in the basil leaves and 1 teaspoon of the balsamic vinegar. Taste and add an additional ½ to 1 teaspoon balsamic vinegar and/or black pepper, if desired. Serve hot. Enjoy as is or with a slice of crusty bread. Bon appetit!