Place potatoes into a glass bowl, coat lightly with about 2 tablespoons of extra-virgin olive oil. Mix to coat.
Trim the ends of the zucchini, then cut lengthwise into 1/4-inch-thick slices. Put in a separate bowl. Top with 3 tablespoons of extra-virgin olive oil, and season with salt and pepper, to taste.
Preheat the oven to 375 degrees F. Lightly oil a deep 2 1/2 to 3-quart baking dish (about 9 by 12 inches). Then place about 1/4 cup of tomato puree and spread out to coat the bottom of the pan.
In zucchini and potato bowls, add 2 tablespoons of garlic each and mix in well with a rubber spatula. Using the potatoes and zucchinis, make a layer in the prepared baking dish, alternating between the potato and zucchini, filling in any gaps with small pieces. Spread 1/2 cup of the tomato puree on top, then sprinkle 2 tablespoons of the Parmesan and 2 tablespoons of the bread crumbs. Top with some of the basil. Repeat the layering of the vegetables, tomato puree, Parmesan, bread crumbs and basil using all the vegetables. Cover the top with the remaining bread crumbs and Parmesan.
Bake, uncovered, for about 1/2 an hour, then cover for the next 45 minutes of baking, finally uncovering for the last 15 minutes, until tender and bubbling, about 1 1/2 hours total cooking time. Remove from the oven and let cool to room temperature to allow the dish to settle and deepen in flavor.
Tiella may be served hot, warm, or at room temperature. Reheat in a low oven, if desired. To serve, cut into squares and carefully lift out of the pan with a spatula.
Bring 2 large pots of water to a boil. Lower the tomatoes into the water, blanch for 45 seconds to 1 minute. Place blanched tomatoes on a baking sheet, and peel the loosened skins. With a colander over a bowl, squeeze the seeds and innards out of each tomato. Set aside and refrigerate the clear strained liquid for other uses. Slice the peeled tomatoes into 1-inch chunks, and place chunks into blender. Puree thoroughly. Pour into bowl.