Preheat oven to 375 degrees F (190 degrees C). Grease an 8x6-inch baking dish with 1 teaspoon olive oil.
Bring a large pot of lightly salted water to a boil. Cook lasagna in the boiling water, stirring occasionally until cooked through but firm to the bite, about 8 minutes. Drain.
Bring another large pot of lightly salted water to a boil. Cook corn in the boiling water until cooked through, about 5 minutes. Cut kernels from the cobs and discard cobs.
Heat 1 1/2 teaspoons extra virgin olive oil in a large skillet over medium-high heat. Saute zucchini ribbons with a pinch of salt and black pepper in hot oil until golden and cooked through, about 5 minutes.
Heat remaining extra virgin olive oil in a large saucepan over medium heat. Cook and stir onion in hot oil until translucent, 5 to 7 minutes. Add garlic; cook and stir until fragrant, about 1 minute. Pour 1/3 of the marinara sauce into the onion mixture; bring to a simmer, reduce heat to low, stir in cream, and cook until sauce is slightly reduced, 5 to 10 minutes.
Pour 1/3 of the marinara sauce into the bottom of the prepared baking dish. Place a layer of lasagna noodles over the marinara sauce, trimming noodles as necessary to fit the baking dish. Spread zucchini ribbons over the noodle layer and top with another layer of lasagna noodles.
Spread ricotta cheese over the lasagna noodles and place another layer of noodles atop the cheese. Pour marinara-cream mixture over the noodles and top with corn kernels.
Place a layer of lasagna noodles over the corn kernels and top noodles with spinach. Add a final layer of noodles and top noodles with remaining marinara sauce. Sprinkle mozzarella cheese and Parmigiano-Reggiano cheese over the top. Cover the baking dish with aluminum foil.
Bake in the preheated oven for 30 minutes. Remove the aluminum foil and continue baking lasagna until cheese is browned and beginning to bubble, about 15 minutes more.