Preheat oven to 375F. In a single layer, arrange eggplant and zucchini on three sheet pans lined with parchment paper. Generously brush them with olive oil. Sprinkle with oregano, salt, and pepper. Bake for 25minutes. Sprinkle garlic over the vegetables and bake for 5 minutes.
When the vegetables are cooked, remove them from the oven. Reduce the oven temperature to 350F.
Bring hot water in a large bowl with a temperature of 140F.Place noodles in hot water one at a time and soak for 15 minutes. Drain and set aside.
Mix ricotta, goat cheese, ½ cup of Parmesan, eggs, basil, 1½ teaspoons salt, and ¾ teaspoon pepper in an electric mixer with a paddle attachment.
In a 9x13x2 inch baking dish, spread 1 cup of marinara sauce. Arrange a third of vegetables, and then a layer of noodles, a third of mozzarella, and a third of ricotta mixture. Repeat this process twice.
Arrange the baking dish on a sheet pan lined with parchment paper. Bake for 1 hour until lasagna is browned.
Allow to rest for few minutes and serve hot.