For the dough: Combine the flour and salt in a large bowl. Make a well in the center of the flour, then pour the olive oil and about 1 cup of lukewarm water. Using a wooden spoon, stir the ingredients until they combine to form a dough, adding a little lukewarm water if necessary. Knead the dough on a lightly floured work surface for 10 minutes. Let rest for 10 minutes.
In the meantime, in a bowl, pour extra-olive oil over the ricotta. Sprinkle with salt, pepper and parsley. Mix until soft. Add more oil, if necessary.
Divide the dough into 3 equal portions. Using the widest opening of a manual pasta rolling machine, roll each portion into a piece 3 feet long, about 4 inches wide and 1/16th of an inch thick. If you do not have a pasta roller, you can also use a lightly floured rolling pin. It is easiest to roll and stuff one portion at a time, rather than rolling all three before stuffing.
For each strip of dough, place rounded teaspoons of ricotta 1-inch from the edge of the lengthwise side of dough, spacing the ricotta about 2 inches apart. For the first strip of dough, sprinkle the ricotta mounds with the freshly grated Pecorino. With the second strip, sprinkle with the chopped mint. Grate the bottarga evenly over the third. Fold the dough in half lengthwise, covering the ricotta-filled portion to the edge, as if you were folding a blanket. Using your fingers, press on the edges of the dough to seal the outside edge. Press also in-between the mounds of ricotta to create individual ricotta portions. Using a sharp knife or a serrated pastry cutter, cut around each mound of ricotta, making 2-inch squares. Place the panzerotti on a lightly floured baking sheet, uncovered. Make the remaining doughnuts, sprinkling the mint over the second batch and grating the bottarga over the third. They can rest for 1 hour.
Fill a deep pot no more than halfway with extra-virgin olive oil. Heat oil over medium-high heat until it reaches a temperature of 350 degrees F. The oil should remain at or around this temperature throughout the cooking process. Working in batches of 6, fry the panzerotti until golden brown. Use a slotted spoon or spider to gently drop the doughnuts into the water, being careful not to splatter the hot oil. Using the same spoon or spider, remove the finished doughnuts to a plate lined with paper towels. Drain on paper towels as they cook. Serve immediately.