In a small saucepan, combine the vinegar with the sugar and boil over moderately high heat until reduced to 3 tablespoons, about 5 minutes. Let cool.

In a large nonstick skillet, heat 3 tablespoons of the olive oil until shimmering. Add the diced eggplant, season with salt and pepper and cook over moderately high heat, stirring, for 2 minutes. Reduce the heat to low and cook, stirring occasionally, until the eggplant is tender, about 10 minutes. Transfer the eggplant to a large bowl. Wipe out the skillet.

Add the remaining 2 tablespoons of olive oil to the skillet and heat until shimmering. Add the diced zucchini and yellow squash, season with salt and pepper and cook over moderate heat, stirring occasionally, until tender, about 7 minutes. Using a slotted spoon, transfer the squash to the bowl with the eggplant.

Add the garlic, celery, and onion to the skillet and cook over moderate heat, stirring occasionally, until softened slightly, about 5 minutes. Add the red bell pepper and cook until all of the vegetables are tender, about 4 minutes longer. Transfer to the bowl with the eggplant and squash.

Add the tomatoes, olives, parsley, oregano, and balsamic syrup to the bowl and fold gently. Season the caponata with salt and pepper. Transfer to a clean bowl. Garnish with the Asiago cheese and serve with the baguette slices.

The caponata can be refrigerated for up to 3 days.