Preheat the oven to 450°F. Preheat a gas or charcoal grill, to medium-high if you have that option (or plan to use a grill pan). Cut each zucchini lengthwise into four 1⁄4-inch- thick strips.
Put the ricotta, mozzarella, Parmesan, basil, egg, and 1⁄2 teaspoon of the salt in a large bowl and mix well. Cover and refrigerate.
If using a grill pan, preheat it over medium-high heat. Mix the oil, the remaining 1⁄2 teaspoon salt, and the pepper on a plate. Brush the mixture over both sides of each zucchini strip, resting the strips on the plate until all are coated.
Arrange the zucchini on the grill or grill pan and cook until the strips are soft but not mushy, about 5 minutes, turning once (stop cooking them as soon as you are able to bend them at a 90-degree angle without breaking). Return them to the plate and set aside to cool slightly.
Remove the cheese mixture from the refrigerator. Mix the tomato sauce with the mint and lemon zest in a small bowl. Spread 1⁄2 cup of this mixture in the bottom of a medium baking dish (11 × 7-inch or equivalent).
Make the involtini one at a time: Roll a bit of the cheese mixture between your palms —you want it to be about the size of a golf ball—and place it on one end of a zucchini strip. Roll the strip up over the lling, rolling it all the way to the end, and place it in the baking dish. When all 8 involtini are arranged in the dish, spread the rest of the tomato sauce over them and place the dish in the oven
Bake until the sauce is bubbling and the cheese is beginning to brown, about 40 minutes. Serve hot!