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 flat on parchment paper or a cooling rack to cool.
Make the béchamel: melt 5 tablespoons butter in a medium saucepan over medium heat. Whisk in 1/3 cup flour and cook 1 minute. Gradually whisk in 4 cups milk, about 1/2 cup at a time, then bring to a gentle boil. Reduce heat and simmer, whisking frequently, until thickened, about 8–10 minutes. Season with a pinch of nutmeg and kosher salt to taste; remove from heat.
Prepare the filling: in a medium bowl combine 2 cups ricotta, 1 large egg, 1 minced garlic clove, 10 ounces thawed and well-drained spinach, 1/2 cup basil pesto, 1/4 cup Parmesan, 1/8 teaspoon crushed red pepper flakes, a pinch of nutmeg, and salt and pepper to taste. Stir until evenly mixed.
Spray a 9x13-inch baking dish with cooking spray and spread 1/2 cup of the béchamel evenly across the bottom.
Assemble roll ups: spread about 1/4 cup of the ricotta-spinach mixture evenly over each lasagna noodle. Top each with 2 tablespoons shredded mozzarella and about 2 tablespoons béchamel. Carefully roll each noodle and place seam-side down in the prepared baking dish.
Top the rolls with the remaining béchamel and the remaining mozzarella and remaining 3 tablespoons Parmesan. Cover the dish with a large sheet of foil sprayed with cooking spray.
Bake covered for 30 minutes at 350°F. Remove the foil and bake an additional 5 minutes, or until the cheese is melted and slightly golden. Let rest 5 minutes before serving.