Step 1

In a large bowl, mix the flour with 3/4 teaspoon of salt. Make a well in the center and add the water, 1/4 cup of the olive oil and the vinegar. Mix until the dough comes together; it will be sticky.

Step 2

Transfer the dough to a floured work surface and knead until smooth and no longer tacky, using a pastry scraper to free it from the work surface. Wrap the dough in plastic wrap; refrigerate for 1 hour.

Step 3

In a large skillet, heat the remaining 2 tablespoons of olive oil. Add the scallions and cook over moderate heat until softened, 3 minutes. Add the greens and cook over high heat, stirring, until wilted, 2 minutes. Stir in the dill, mint and oregano and cook until wilted, 1 minute. Transfer the contents of the skillet to a colander and let cool. When the greens are cool enough to handle, squeeze out any excess liquid.

Step 4

In a large bowl, combine the greens with the feta, Parmigiano-Reggiano cheese and the egg. Season the filling with 1/4 teaspoon each of salt and pepper.

Step 5

Lightly flour a large, rimmed baking sheet. Turn the dough out onto a lightly floured work surface; cut into 4 equal pieces. Roll each piece out to a 9-inch round 1/16 inch thick. Using an 8-inch plate as a template, cut an 8-inch disk from each round. Mound one-fourth of the filling on the lower half of each disk. Fold the dough over to make a half-moon; press the edge of the dough to seal. Using a lightly floured fork, crimp the edges. Transfer the hand pies to the prepared baking sheet.

Step 6

In a large skillet, heat 1/4 inch of the vegetable oil over moderately high heat until shimmering. Add 2 pies at a time and fry, turning once, until deep golden brown, about 6 minutes. Drain the pies on paper towels. Serve hot.