Boil the potatoes in their skins until tender. Let them cool slightly, then peel and mash them thoroughly using a potato ricer or masher until completely smooth.
In a large bowl, combine the mashed potatoes with the flour, egg yolk, and salt. Gently knead the mixture until a soft, non-sticky dough forms. Be careful not to overwork the dough, as this will result in tough gnocchi.
Divide the dough into four equal portions. On a lightly floured surface, roll each portion into a long rope about 2 cm (¾ inch) in diameter. Using a sharp knife, cut the ropes into 2 cm (¾ inch) pieces.
Shape each piece by gently pressing it against the tines of a fork to create ridges. This helps the sauce cling to the gnocchi.
Bring a large pot of salted water to a boil. Add the gnocchi and cook until they float to the surface (about 2-3 minutes). Drain well.
In a large skillet, heat the olive oil over medium heat. Add the finely chopped garlic and sauté until fragrant, about 1 minute.
Pour in the tomato sauce, season with freshly ground black pepper and a pinch of salt. Simmer for 5-7 minutes, allowing the flavors to meld.
Add the cooked gnocchi to the tomato sauce and gently toss to coat each piece evenly.
Stir in the roughly torn fresh basil leaves and the diced mozzarella. Gently mix until the cheese begins to melt.
Serve immediately, garnished with freshly grated Parmesan cheese and extra basil leaves.
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