Preheat oven to 350°F (175°C). Line a baking sheet with parchment paper or a silicone mat, or spray it with nonstick baking spray.
Make the dough: on a clean work surface mound the 1 cup (125 g) flour and make a well in the center. Break the 2 eggs (for the dough) into the well, add the 2 tablespoons cold water and the 1 teaspoon vegetable oil. Use a fork to mix the wet ingredients, then gradually incorporate the flour until a shaggy mass forms.
Knead the dough by hand until it becomes soft, smooth, and elastic. If the dough is too sticky, add a little more flour; if it is too dry, add a small bit of water. Cover the dough and set it aside.
Make the filling: in a medium bowl combine the 2 eggs (beaten), the 12 ounces (340.2 g) grated Romano cheese, and freshly ground black pepper to taste. Stir until thoroughly combined.
Lightly flour your work surface and roll the dough to about 1/8–1/4 inch thickness. Lay the rolled dough flat and place 1 tablespoon portions of the filling about 2 inches apart across one half of the dough.
Fold the unused half of the dough over the filling. Press around each mound to release any trapped air and to seal. Cut into squares with a knife or pasta roller. If you cut with a knife, press the edges with a fork to help seal each piece.
Transfer the sealed pigu to the prepared baking sheet, spacing them so they are not touching.
Make the egg wash by beating together 1 egg and 1 tablespoon water; brush the tops of the pigu with the egg wash. Bake for 10–15 minutes, or until the tops are lightly browned.