Boil the potatoes in a large pot of salted water until fork-tender and skin easily peels away, about 20 minutes. Drain and let them cool slightly.
Peel the potatoes while still warm and mash them thoroughly using a fork or potato ricer until smooth and free of lumps.
Spread the mashed potatoes out on a clean work surface and let them cool slightly, then create a well in the center.
Add the beaten egg and salt into the well, then gently mix to combine.
Gradually incorporate the flour into the potato mixture, kneading gently until a soft, slightly sticky dough forms.
Divide the dough into 4 equal portions and roll each into a long rope about ¾ inch thick.
Cut each rope into 1-inch pieces, then gently roll each piece over a fork or a gnocchi board to create ridges, if desired.
Bring a large pot of salted water to a boil. Carefully add the gnocchi and cook until they float to the surface, about 2-3 minutes. Use a slotted spoon to remove them.
Heat olive oil in a large skillet over medium heat. Add the chopped sun-dried tomatoes and cook until fragrant, about 2 minutes.
Add the cooked gnocchi to the skillet and fry until golden brown and crispy at the edges, about 4-6 minutes.
Stir in the minced garlic and chopped basil, cooking just until fragrant, about 1 minute. Toss to coat evenly.
Serve the gnocchi hot, garnished with additional basil if desired, and enjoy the rustic, flavorful bites with crispy edges and tender centers.