Preheat the oven to 350°F (175°C).
Cook the lasagna noodles according to package directions until al dente, drain, and set aside.
Heat the vegetable oil in a large skillet over medium heat. Add the chopped onion and garlic and cook until softened, about 3–5 minutes.
Add the ground beef and brown it, breaking it up with a spoon, until no pink remains.
Stir in the canned tomatoes, tomato sauce, salt, pepper, basil, oregano, and sugar; bring to a simmer and cook uncovered for about 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. Spoon half of the meat sauce into the bottom of the dish.
Place 3 cooked lasagna noodles over the sauce, spread half of the cottage cheese mixture on the noodles, and top with half of the sliced mozzarella.
Repeat the layers: remaining meat sauce, 3 noodles, remaining cottage cheese mixture, and remaining mozzarella. Sprinkle the Parmesan cheese evenly over the top.
Bake the assembled lasagna in the preheated oven for 30–40 minutes, until bubbly and the cheese is melted. Let rest a few minutes before slicing and serving.
