Preheat the oven to 450°.
Wash and dry okra, summer squash, and eggplants. Leave okra whole and cut summer squash and eggplants into ¼-inch slices. Mix cane vinegar and sorghum syrup with 3 tablespoons olive oil and mix. Toss the eggplant slices in the vinegar-syrup mixture. In separate bowls, toss okra and summer squash each with 3 tablespoons olive oil. Season all vegetables with salt and pepper to taste.
Place the eggplant slices on a baking sheet lined with parchment paper or greased with olive oil. Roast until golden brown and tender, about 20 minutes.
Roast okra until just brown and tender, about 15 minutes, turning once or twice. Let cool and slice okra lengthwise.
Place the summer squash slices on a baking sheet lined with parchment paper or greased with olive oil. Cook until tender and slightly brown, about 10 minutes.
Assembling and baking the Lasagna
Pre-heat the oven to 375°.
Oil a 9- by 13-inch baking dish.
Spread a ladle (about ½ cup) of sauce on the bottom of the pan. Cover with lasagna noodles.
Add a layer eggplant. Spread another ladle of sauce over the eggplant. Layer with slices of mozzarella and sprinkle generously with grated Parmesan. Cover with lasagna noodles. Repeat for summer squash and okra. Finish the top layer with lasagna noodles, cover with tomato sauce, top with mozzarella and enough grated Parmesan to cover the mozzarella.
Cover with a piece of oiled aluminum foil. (If you want to make ahead, it can now be refrigerated overnight.)
Bake 25 minutes (30 if refrigerated). Remove the foil and continue baking until the top turns golden brown, about 15 minutes. Remove from the oven and let it rest for at least 5 minutes. Cut and serve.
Tip: Prior to assembling the lasagna, divide the sliced mozzarella into four portions. The last one should bigger. This ensures you have enough mozzarella for each interior layer and to cover the top.