Preheat oven to 350◦. Lightly spray a 9 x 13-inch baking dish with nonstick cooking spray.
Heat the olive oil in a large skillet over medium heat. Add the shallots and sauté until translucent, about 4 minutes. Add the garlic and cook for another 30 seconds or until fragrant. Add the spinach by handfuls, letting it wilt in between each. Place the mixture in a colander to drain.
Combine the ricotta, mozzarella, basil, salt, and 1 cup of the Parmesan.
Spread 1/2 cup marinara sauce in the bottom of the baking dish. Top with 3 noodles, trimming them as needed to fit. Spread half of the spinach mixture over the noodles, then half of the ricotta mixture, followed by 1 cup of marinara. Repeat with 3 more noodles, the remaining spinach and ricotta mixtures, and another cup of marinara. Finish with the final 3 noodles and 1 1/2 cups of marinara.
Bake the lasagna for 1 hour, then sprinkle the top with the remaining 1 cup of Parmesan and continue baking until the cheese is melted, about 15 minutes. Let the lasagna stand for 30 minutes before cutting and serving.