Preheat the oven to 350°F (175°C) and lightly spray a 9x13-inch baking pan with nonstick spray; set aside.
Bring a large pot of salted water to a boil and cook the jumbo pasta shells until al dente according to package directions, about 8–9 minutes.
Drain the shells in a colander and transfer them to a baking sheet, placing them upside down to dry and prevent sticking.
Meanwhile, heat a large skillet over medium heat and brown the ground chuck until no pink remains, about 5–7 minutes; drain excess fat.
Lower the heat to medium-low, add the olive oil and diced onion to the skillet with the beef, and cook until the onion is translucent, about 3 minutes; add the minced garlic and cook 30 seconds more.
Reduce heat to low, stir in the marinara sauce, and simmer uncovered for about 30 minutes, stirring occasionally.
In a medium bowl, combine the beaten egg, cottage cheese, 1 cup of the shredded mozzarella, and 1/2 cup of the grated Parmesan until evenly mixed to make the filling.
Spoon about three-quarters of the meat sauce into the bottom of the prepared baking dish, spreading it evenly.
Using a rounded tablespoon, stuff each shell with the cheese mixture and arrange the filled shells open-side up on top of the sauce in the baking dish.
Spoon the remaining meat sauce between the shells (avoid covering the tops of the shells directly).
Cover the baking dish with aluminum foil and bake for 30 minutes. Remove the foil, sprinkle the reserved 1 cup mozzarella and 1/4 cup Parmesan over the shells, and bake 5–7 minutes more until the cheese is melted and bubbly.
Garnish with chopped fresh basil, if using, and serve warm.
