Preheat the oven to 350°F (175°C). Spread 1 teaspoon olive oil over the bottom of a 9x13-inch baking dish and set aside.
Cook the pasta shells in salted boiling water until al dente according to package directions. Drain and set aside to cool slightly so they are easy to handle.
Heat 2 teaspoons olive oil in a skillet over medium heat. Add the diced onion and minced garlic and cook until softened, about 3 minutes.
Add the ground beef to the skillet and cook until browned, breaking it up with a spoon; drain any excess fat.
Stir the Greek yogurt, cream cheese, and 1 cup shredded Parmesan into the cooked meat mixture until evenly combined to make the filling.
While the filling is mixing, make the sauce: combine the crushed tomatoes, tomato paste, dried basil, dried oregano, and the 1 teaspoon kosher salt in a saucepan and simmer until heated through.
Spread 1 cup of the tomato sauce over the bottom of the prepared baking dish.
Spoon the meat-and-cheese filling into each cooked shell and arrange the filled shells in the baking dish in a single layer.
Spoon the remaining tomato sauce over the stuffed shells, sprinkle the remaining 1/2 cup Parmesan and the cup of grated mozzarella over the top, cover with foil, and bake for 20 minutes. Remove the foil and bake 10 minutes more until bubbly.
Remove from the oven and let rest a few minutes before serving.
