Bring a large pot of salted water to boil. Position an oven rack in the middle of the oven and preheat to broil. Rub 1 teaspoon olive oil on the bottom of a baking sheet to prevent the noodles from sticking.
Cook the lasagna according to the package directions for al dente.
Meanwhile, heat 1 tablespoon olive oil in an oven-safe medium skillet over medium-high heat. Add the sausage and cook until sausage is browned, breaking it into crumbles with a wooden spoon, 4 to 5 minutes. Set aside to cool. Reserve the skillet for assembling the lasagna.
Combine the ricotta, spinach, mozzarella and 2 tablespoons grated Parmesan in a medium bowl.
When the lasagna noodles finish cooking, drain them, then rinse them with cold water. Lay the noodles on the prepared baking sheet in a single layer so the edges touch each other. Cut the noodles in half lengthwise with the tip of a paring knife.
Spread the ricotta mixture evenly over the noodles with an offset spatula. Evenly sprinkle the sausage on top of the ricotta mixture.
Spread the marinara sauce on the bottom of the medium skillet used to cook the sausage. Carefully roll each noodle into a pinwheel starting from the short end, using your fingers to keep the ricotta and sausage contained. Rotate the pinwheels so the frilling ends are on top, then transfer them to the prepared skillet with the marinara sauce in a snug, even layer.
Brush the tops of the lasagna roll ups with the remaining 2 tablespoons of the olive oil and sprinkle evenly with the remaining 1/4 cup grated Parmesan. Put the skillet of over medium heat and cook until the sauce simmers and the filling is warmed through, 2 minutes. Broil until the top is dark golden, 2 to 3 minutes.