1. Preheat oven to 350 degrees F. Line a baking sheet with parchment paper or silicone mat, or spray with a non-stick baking spray.
2. Prepare the dough as you would a typical pasta dough, with a mound of flour with a well in the center. Into the middle break the eggs, add the water and oil, and mix together with a fork until combined and then begin incorporating the flour a little at a time. Once the dough comes together as a shaggy mass, knead it by hand until you have a soft, smooth, elastic dough. Add more flour or water, a small bit at a time, until that consistency is reached. Cover and set aside.
3. To make the filling, combine the eggs with the Romano cheese and black pepper in a medium bowl and stir until thoroughly combined.
4. Roll out the pasta dough to about 1/8-inch to 1/4-inch thickness. Place 1 tablespoon of the filling two inches apart across half of the rolled-out dough. When finished, fold the unused half the dough over the filling. Press around each mound of filling to release air and cut into squares with a knife or pasta roller. If using a knife, seal the edges with a fork. Place pigu on the baking sheet.
5. With a fork, beat together the egg and water and brush the tops of the pigu with the egg wash.
6. Bake for 10-15 minutes or until lightly browned.