Slice the eggplant lengthwise into 1/8- to 1/4-inch-thick slices. Sprinkle both sides generously with salt, then arrange the eggplant slices in a single layer between several layers of paper towels. Set aside to draw out the moisture, about 30 to 40 minutes. Pat very dry with the paper towels.

Meanwhile, bring a large pot of salted water to a boil. Add the lasagna noodles and cook as the label directs for al dente. When done, place them in a single layer on sheets of parchment paper to prevent sticking.

Heat the olive oil in a large nonstick skillet over medium-high heat. Add 4 to 5 eggplant slices and cook until golden brown and crisp, 1 to 2 minutes per side. As you proceed, you may want to place the finished slices on a rack over a pan or on some paper towels to absorb the excess oil. Repeat with the remaining eggplant, adding more oil to the skillet if needed.

Preheat the oven to 350 ̊ F.

Pour about a cup of the marinara sauce into an 8-inch square baking dish.

Spread several tablespoons of marinara sauce on a lasagna noodle; top with 1 tablespoon Parmesan, 3 heaping tablespoons mozzarella, about a quarter-ounce of baby arugula then a sprinkling of basil. Top that off with another lasagna noodle, then 2 eggplant slices to cover the length of the entire noodle. Top the eggplant with 2 heaping tablespoons of the tomato sauce. Starting with a short end, roll up the stack into a spiral.

Use a sharp knife to slice the lasagna roll in half to make 2 spirals. Place each piece in the baking dish cut-side down with the wavy edges of the noodle facing up. Continue with the remaining lasagna noodles and eggplant slices until you have 16 spirals in the baking dish. Top with any remaining Parmesan and mozzarella. 

Cover the dish with foil and bake for 30 minutes, until the cheese is melted and the sauce is bubbling. Uncover and continue to bake until browned on top, for 15 more minutes. Top with basil and serve with additional tomato sauce.