|Toss the eggplant cubes with a teaspoon or so of salt. Set the cubes in a colander to drain for about 20 minutes.Heat 2 tablespoons of olive oil in a heavy-bottomed pot. Pat the eggplant dry, add to the pan, and cook over medium heat, stirring frequently, until golden. Add a bit more oil if the eggplant absorbs all the oil and sticks to the bottom of the pan. Remove the eggplant when done and set aside.
In the same pot, pour in 2 more tablespoons olive oil. Add onions and cook for about 7 minutes, or until soft and translucent. Add the garlic, basil bouquet, dried chile flakes, and a bit more salt.
Cook for 2 or 3 minutes, then stir in peppers. Cook for a few more minutes, then stir in summer squash. Cook for a few more minutes, then stir in tomatoes.
Cook for 10 minutes longer, then stir in eggplant and cook for 10 to 15 minutes more, until all the vegetables are soft. Remove the bouquet of basil, pressing on it to extract all its flavors, and adjust the seasoning with salt.
Stir in the chopped basil leaves and more extra virgin olive oil, to taste. Serve warm or cold.