Combine eggplant and salt in a large bowl. Transfer to a colander, place a plate on the eggplant and weigh down the plate with cans. Set the colander in the sink to drain for 1 hour.
Meanwhile, heat 1/4 cup oil in a large skillet over medium heat. Add bay leaves and let sizzle for about 1 minute to flavor the oil. Stir in onions and celery. Reduce heat to medium-low and cook, stirring occasionally, until the vegetables are soft, about 30 minutes; do not let them brown. Add tomatoes and capers, increase heat to medium and cook; stirring occasionally, just until the tomatoes start to break down, about 5 minutes. Stir in olives; transfer everything to a large bowl.
When the eggplant is ready, rinse under running water to get rid of as much salt as possible. Dry thorougly on paper towels.
Heat the remaining 1/4 cup oil in the same skillet over medium heat until very hot. Add the eggplant and cook, tossing and stirring, until softened, 10-15 minutes. Add chiles and cook, stirring, until they are softened, 5-10 minutes. Transfer the eggplant and peppers to the bowl with the tomato mixture and gently stir to combine.
Whisk vinegar and honey in a small saucepan; bring to a boil over medium heat. Simmer until thickened and reduced to about 1/4 cup, about 5 minutes. Stir into the vegetables along with parsley and basil. Serve at room temperature.