In a large mixing bowl, combine the flour and salt. Make a well in the center and add the eggs and olive oil.
Using a fork or a dough mixer, gradually incorporate the flour into the eggs and oil, mixing until a dough begins to form.
Knead the dough on a floured surface for about 10 minutes, until it's smooth and elastic. Wrap in plastic wrap and let it rest for 30 minutes at room temperature.
Boil the diced potatoes in salted water until tender, about 15 minutes. Drain and mash them while still warm.
Stir in the grated Pecorino cheese and chopped mint. Season with salt and pepper to taste. Set aside to cool.
Roll out the rested pasta dough into a thin sheet. Using a round cutter, cut out circles approximately 3 inches (7.5cm) in diameter.
Place a spoonful of the potato filling in the center of each dough circle.
To seal, fold the dough over the filling to create a half-moon shape. Pinch the edges to seal, then roll and press the edge to create a braided look.
Bring a large pot of salted water to a boil. Gently drop the Culurgiones in and cook for about 4-5 minutes, or until they float to the surface. Remove with a slotted spoon and drain.
In a skillet or saucepan, heat the olive oil over medium heat. Add the chopped onion and minced garlic, sautéing until soft and fragrant.
Pour in the crushed tomatoes, sugar, and season with salt and pepper. Simmer for 20 minutes, stirring occasionally.
Serve the boiled Culurgiones topped with the warm tomato sauce. Garnish with fresh basil leaves.