Preheat grill on high or oven to 400 degrees. If doing this in your oven, adjust one oven rack to upper-middle position and second oven rack to lowest position.
Meanwhile, halve each zucchini lengthwise. With a small spoon, scoop out the seeds and most of the flesh so that the walls of the zucchini are about 1/4-inch thick all the way around. Season the cut sides of the zucchini with salt and pepper and brush them lightly with 1 tablespoon of the oil. Set the zucchini halves cut-side down on a baking sheet or directly on your grill. Meanwhile, toss the potatoes with diced zucchini meat that you scooped out along with about 2 tablespoons of the olive oil, salt, and pepper, to taste, in a small bowl and spread in a single layer on a baking sheet or grill pan. Roast the zucchini halves until slightly softened and skins are slightly wrinkled, about 10 minutes while simultaneously roasting the potatoes until they are tender and lightly browned, 12 to 14 minutes. When the zucchini has finished roasting/grilling, remove it from the oven and using tongs, flip the zucchini halves over on the baking sheet and set aside.
While the zucchini and potatoes are roasting, heat the remaining oil in a 12-inch skillet over medium heat until shimmering, about 2 minutes. Add the onion and cook, stirring occasionally, until softened and beginning to brown, about 10 minutes. Stir in the garlic and cook until fragrant, about 30 seconds. Add the tomatoes and cooked potatoes and zucchini meat; cook, stirring occasionally, until heated through, about 3 minutes. Off the heat, stir in the basil, 1 cup cheese,dried Italian spices and salt and pepper to taste.
Divide filling evenly among the zucchini halves on the baking sheet or grill pan, spooning about 1/2 cup into each, and packing it; sprinkle with remaining cheese. Return the baking sheet to the oven or grill, this time to the upper rack of oven or closed grill, and bake the zucchini until heated through and cheese is spotty brown, about 6 minutes. Serve immediately.