Place an oven rack in the middle position and preheat the oven to 375°F (190°C).
Make the sauce: heat olive oil in a medium saucepan over medium heat and sauté the chopped onion until soft and fragrant, about 5 minutes.
Add the minced garlic, red pepper flakes, and Italian spice mix, then stir in the tomato paste and cook 1 minute more.
Pour in the crushed tomatoes, bring to a simmer, and cook gently for about 20 minutes. Season with salt and pepper to taste.
Prepare the filling: reserve 1 cup of the shredded cheese for topping, then in a large bowl combine the ricotta, remaining 3 cups of shredded cheese, lightly 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 pat dry on a clean kitchen towel.
Assemble the casserole: spread half of the sauce evenly on the bottom of a 9 x 13-inch baking dish.
Place about 1/4 cup of the cheese and spinach filling along the short edge of each softened noodle and roll up tightly; place each roll seam-side down over the sauce in the baking dish.
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 shredded cheese over the top, and bake uncovered until the cheese is melted, about 5 more minutes.
Let the casserole rest for 10 minutes before serving.