1.To make pasta, place flour in a large bowl and create a well in the centre. In a separate bowl, place eggs, oil and a large pinch of salt. Beat lightly. Pour egg mixture into well of flour. Lightly mix together. Knead for about 2-3 minutes, or until dough is smooth. Cover with a tea towel and set aside to rest for 30 minutes before rolling. (Alternatively, make dough the day before and refrigerate overnight, wrapped in plastic wrap. Bring to room temperature 1 hour before rolling.)
2.To make filling, boil potatoes in plenty of salted water. Drain and push through a ricer, then return to saucepan. Add oil and beat lightly to mix together. Stir in garlic, pecorino and mint leaves. Set aside.
3.Roll out pasta, using second-thinnest setting on a pasta roller. Lay sheet of pasta on a work surface. Cut into long, 6cm-wide strips.
4.Dollop rounded teaspoons of potato filling about 1.5cm apart onto pasta, 1cm away from the edge. Lightly brush edges of pasta with water, fold pasta over filling and press dough together to seal. Pinch out all the air, then use a fluted pastry cutter to cut out dumplings. Repeat until all dough and filling mixture is used.
5.Bring large pot of salted water to boil, add dumplings and cook for 2-3 minutes. Drain and serve with grated pecorino, tomato compote and a drizzle of olive oil.