Slice off the stem ends of the zucchini. Using a mandoline, slice the zucchini into long ¼-inch-thick planks. Place the zucchini slices in single layers on 2 paper-towel-lined baking sheets and generously sprinkle salt on both sides. Let sit for 10 minutes to allow the zucchini to release excess water.

Meanwhile, combine crushed tomatoes, halved cherry tomatoes, garlic, basil and ½ teaspoon each salt and pepper in a bowl. In a separate bowl, combine the ricotta, mint, 1 tablespoon olive oil and ½ teaspoon each salt and pepper. Set both bowls aside.

Heat oven to 425 degrees.

Discard the damp paper towels from the baking sheets and pat the zucchini dry. Drizzle 1 tablespoon of olive oil over each baking sheet of zucchini and turn to coat the pieces on all sides. Place a grill pan or cast-iron skillet over medium heat. Once the pan is hot, cook the zucchini in single-layer batches for 3 to 5 minutes per side, until they are charred and tender. Transfer cooked zucchini to one of the sheet pans until it is all cooked.

Assemble the lasagna: Spread 1 cup of tomato sauce across the bottom of a 9-by-13-inch baking pan, followed by a single layer of grilled zucchini. Next, layer half each of the grated Parmesan, mozzarella and ricotta in dollops. Add another cup of sauce, a layer of zucchini and the rest of the three cheeses. Layer any remaining zucchini planks across the top along with the rest of the sauce.

Cover with foil and bake for 30 minutes, then uncover and broil for 3 to 4 minutes, until the top is as charred as you like. Let sit for 10 minutes and then slice and serve.