Preheat the oven to 350°F (175°C).
Bring a large pot of lightly salted water to a boil. Add the jumbo shells and cook 8–10 minutes until al dente; drain and let cool slightly so they are easy to handle.
In a large bowl, combine the beaten egg, ricotta, half of the shredded mozzarella, half of the grated Parmesan, dried basil, salt, and black pepper until well mixed.
Spoon half of the pasta sauce across the bottom of a 9×13-inch baking dish, spreading it into an even layer.
Stuff each cooked shell with the ricotta mixture (you should have about 18–20 shells) and arrange them in a single layer over the sauce in the baking dish.
In a medium bowl, stir the remaining sauce together with the reserved shredded mozzarella and grated Parmesan, then pour this sauce evenly over the stuffed shells.
Bake uncovered in the preheated oven for about 30 minutes, until the edges are bubbly and the shells are set.