Preheat the oven to 350°F (175°C) and spray a 9×13-inch (3‑quart) baking pan with nonstick spray.
Bring a large pot of salted water to a boil and cook the jumbo shells until al dente according to package directions, about 8–9 minutes; drain and place shells open-side down on a baking sheet to dry.
In a large skillet over medium heat, brown the ground chuck until cooked through, about 5–7 minutes; drain any excess fat.
Reduce heat to medium-low, add the olive oil to the skillet with the beef, then add the diced onion and cook until translucent, about 3 minutes; stir in the minced garlic and cook for 30 seconds.
Reduce heat to low and add the marinara sauce to the skillet; simmer uncovered, stirring occasionally, for about 30 minutes.
While the sauce simmers, combine the beaten egg, cottage cheese, 1 cup mozzarella, and 1/2 cup Parmesan in a medium bowl until evenly mixed.
Pour about three-quarters of the meat sauce into the bottom of the prepared baking dish and spread into an even layer.
Fill each shell with approximately 1 rounded tablespoon of the cheese mixture and arrange the stuffed shells open-side up on top of the sauce in the baking dish.
Spoon the remaining meat sauce between the shells (avoid covering the openings) and cover the dish tightly with aluminum foil.
Bake covered for 30 minutes. Remove the foil, sprinkle the remaining 1 cup mozzarella and 1/4 cup Parmesan over the shells, then bake 5–7 minutes more until the cheese is melted and bubbly.
Garnish with chopped fresh basil if desired and serve.