Heat 2 tablespoons olive oil in a large skillet. Add diced onions, tomatoes, eggplant, and celery. Season well with salt and pepper; cook over medium heat for 10 minutes.
Add the tomato paste, mix well, and cook an additional 5 minutes. Add the pine nuts, black olives, sultanas, and oregano. Cook 1 minute longer, then add the vinegar, mix well, and turn off the heat. Set aside.
Preheat oven to 425°F (220°C).
Add the remaining 3 tablespoons olive oil to a 10-inch cast-iron skillet over medium heat. Sprinkle evenly with the teaspoon sugar. Arrange zucchini and yellow squash slices, overlapping, in a decorative pattern in the bottom of the skillet. Season lightly with salt and pepper.
Make a second layer of squash slices and season lightly with salt and pepper. Cook for 2-3 minutes without disturbing, then spread the eggplant-tomato filling on top. Remove from heat and sprinkle evenly with the Parmigiano-Reggiano.
Roll out the piecrust, and place it on top of the filling and cheese. Trim edges and tuck them in around the perimeter. Using a fork, poke several steam holes in the crust. Bake the tart for 25 minutes or until the crust is golden brown.
Remove from the oven and allow to sit for 5 minutes. If the crust seems stuck to the pan, run a paring knife around the edges. Flip the Caponata Tart over onto a large platter. Sprinkle with additional Parmigiano-Reggiano, slice and serve. The tart can be served hot or at room temperature.