Bring a large pot of salted water to a boil and cook the lasagna noodles according to package instructions until al dente; drain and set aside.
Preheat the oven to 350°F (175°C).
Heat the vegetable oil in a large skillet over medium heat. Add the chopped onion and garlic and cook gently until the onion is softened and translucent, about 3–5 minutes.
Add the ground beef to the skillet and brown, breaking it up with a spoon, until fully cooked and no pink remains.
Stir in the canned tomatoes, tomato sauce, salt, pepper, basil, oregano, and sugar. Bring to a simmer, then cook uncovered for 30 minutes, stirring occasionally.
While the sauce simmers, beat the egg in a bowl and mix in the cottage cheese until combined.
Grease a 2-quart casserole dish. Spread half of the meat sauce on the bottom, arrange 3 lasagna noodles over the sauce, spoon half of the cottage-cheese mixture over the noodles, and top with half of the sliced mozzarella.
Repeat the layers: remaining meat sauce, remaining 3 noodles, remaining cottage-cheese mixture, and remaining mozzarella. Sprinkle the grated Parmesan evenly over the top.
Bake in the preheated oven for 30 to 40 minutes, until bubbly and the cheese is melted. Let rest a few minutes before serving.