Make sure you have all the ingredients before you begin.
Sauce
In a large saucepot over medium heat, warm the oil. Add the garlic and cook until just lightly golden.
Stir in the tomatoes with juice, basil, and salt and pepper to taste. Bring the sauce to a simmer and cook until the sauce thickens and the tomatoes have mostly fallen apart, approximately 30 to 40 minutes. Adjust the heat as needed to keep the sauce at a steady simmer.
Prepare your filling and dough while the sauce is simmering. When finished, set the sauce aside for later. You can reheat it.
Filling
Place the potatoes in a pot and cover them with cold water. Add a pinch of salt to the water. Bring the water to a boil over high heat. Turn the heat down to a rolling simmer until the potatoes are tender when pierced with a fork.
Rinse the potatoes under cold water in a colander and set them aside to drain and cool. Remove the skins once cool enough to handle.
Place the potatoes in a large bowl and mash them. Add the pecorino, ricotta, mint, and garlic. Mix well to combine everything. You can use your hands. Cover the bowl with plastic wrap and place it in the refrigerator for a couple of hours.
Dough
In a large shallow bowl or on a work surface, Slowly add salted water to the flour and knead for several minutes until a smooth ball is formed.
Wrap the dough in plastic wrap and put it aside to rest for 30 minutes.
Assemble and Cook
On a well-floured work surface, roll out the dough to approximately 1/16 inch thickness. Using a glass or biscuit cutter, make 3-inch circles of the dough.
In each of the circles, place a walnut-size portion of filling. Close the pasta to form a football shape with decorative ridges to resemble a wheat stalk, as shown in this video. Pinch the end and fold it onto itself, and repeat it until you close the culurgione. Place them on a floured sheet pan.
Place your sauce in a pot over medium-low heat to warm it up.
In a large pot of salted, boiling water, cook the culurgiones in batches for 4 to 5 minutes. They should start to float to the top in about 2-1/2 minutes. Remove the pasta from the pot with a spider or slotted spoon and place them into a warm pot of sauce just to coat them.