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