Boil the potatoes in their skins until tender. Peel them while still warm, then mash using a potato ricer or masher. Let them cool completely.
Once the potatoes are cooled, add the flour, egg, and a pinch of salt. Knead the dough until it forms a smooth, soft, and non-sticky consistency.
Heat the olive oil in a skillet over medium heat. Sauté the finely chopped onion and minced garlic until softened. Add the ground meat and cook, breaking it up with a spoon, until browned. Season with salt, pepper, and chopped fresh basil.
Roll the gnocchi dough into small balls, flatten them slightly, and place a small spoonful of the meat filling in the center. Carefully enclose the filling with the dough, forming round dumplings.
Bring a large pot of salted water to a boil. Gently drop the gnocchi into the boiling water and cook for 2-3 minutes, or until they float to the surface. Drain well and set aside.
In a large skillet, heat the tomato sauce. Add the cooked gnocchi and gently toss to coat. Serve hot, sprinkled with grated Parmesan cheese.
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