Place eggplant pieces in a large colander over your sink. Sprinkle with salt. Leave for 20 minutes as the eggplant sweats out its bitterness. Pat dry to remove water and excess salt.
Coat a large wide pan with olive oil. Add the onions, season with salt and paprika, and bring the pan to a medium heat.
Cook the onions until they are soft and very aromatic but have no color, 7 to 8 minutes. Add the garlic, tomatoes, zucchini, bell pepper, eggplant, ¾ cup of water, and fresh thyme springs. Stir in black pepper and rosemary. Cook for about 15 minutes, or until the zucchini and bell peppers are soft.
Stir in the basil and taste to make sure the seasoning is correct. Serve warm or at room temperature.