Preheat oven to 375°F (190°C) and place a rack in the middle position.
Make the sauce: Heat the olive oil in a medium saucepan over medium heat. Sauté the chopped onion until soft and fragrant, about 5 minutes.
Add the minced garlic, red pepper flakes, and Italian spice mix to the pan and cook 1 minute, then stir in the tomato paste and cook another minute.
Pour in the crushed tomatoes, stir to combine, and simmer the sauce gently for about 20 minutes. Season with salt and pepper to taste.
Prepare the filling: Reserve 1 cup of the shredded cheese for topping. In a large bowl, combine the ricotta, remaining 3 cups shredded cheese, beaten egg, 1/2 teaspoon salt, 1/2 teaspoon pepper, and the chopped spinach until evenly mixed.
Soak the no-boil lasagna noodles in hot water for about 5 minutes to soften, then drain and lay them on a clean kitchen towel to remove excess water.
Assemble the casserole: Spread half of the sauce evenly across the bottom of a 9" x 13" baking dish.
Place about 1/4 cup of the cheese and spinach filling along the short edge of each noodle, then roll the noodle tightly and place seam-side down in the baking dish over the sauce. Repeat with remaining noodles and filling.
Pour the remaining sauce over the rolled manicotti and cover the dish tightly with foil.
Bake covered for about 40 minutes. Remove the foil, sprinkle the reserved 1 cup of cheese evenly over the top, and bake uncovered until the cheese is melted, about 5 more minutes.
Let the manicotti cool for 10 minutes before serving.