Preheat the oven to 350°F (175°C). Salt a large pot of water and cook the lasagna noodles according to package directions until al dente. Drain and lay noodles flat on parchment paper or a cooling rack to cool.
Make the béchamel: melt the butter in a medium saucepan over medium heat. Whisk in the flour and cook for 1 minute. Gradually whisk in the milk, about ½ cup at a time, until smooth. Bring to a gentle boil, then reduce heat and simmer, whisking frequently, until thickened, about 8–10 minutes. Season with a pinch of nutmeg and salt to taste, then remove from heat.
Prepare the filling: in a medium bowl, combine ricotta, the large egg, minced garlic, thawed and well-drained spinach, basil pesto, ¼ cup grated Parmesan, crushed red pepper flakes, and a pinch of nutmeg. Stir until evenly mixed and season with salt and pepper to taste.
Assemble the pan: spray a 9x13-inch baking dish with cooking spray. Spread ½ cup of the béchamel evenly on the bottom of the dish.
Assemble each roll-up: spread about ¼ cup of the ricotta-spinach mixture evenly over one noodle. Sprinkle 2 tablespoons of shredded mozzarella and drizzle 2 tablespoons of béchamel over the filling. Roll the noodle up tightly and place seam-side down in the prepared baking dish. Repeat with remaining noodles and filling.
Top the rolls with the remaining béchamel, the remaining mozzarella, and the remaining 3 tablespoons Parmesan. Cover the dish with a piece of foil sprayed with cooking spray (to prevent sticking).
Bake covered for 30 minutes. Remove the foil and bake an additional 5 minutes, or until the cheese is melted and lightly browned. Let rest 5 minutes before serving.