Step 1Make the dough: In a large bowl, whisk together the flour, salt and sugar. Add the shortening and incorporate using a pastry cutter or fork (the dough will look like moist sand). Cut in the butter just until it is reduced to pea-sized pieces. Sprinkle the vinegar and 8 tablespoons water over the mixture, and stir together until the ingredients are combined to form a dough. Remove the dough to a lightly floured surface and knead a few times until it comes together in a single mass. If the dough is too crumbly and dry, gently work in additional water, 1 tablespoon at a time. Mold the dough into a disk roughly 8 inches in diameter. Cover the disk tightly with plastic wrap and refrigerate at least 2 hours, preferably overnight.
Step 2Bring a large pot of salted water to a boil. Add the greens to the water and boil until they're tender, 12 to 15 minutes. Drain the greens, then shock in a bowl of ice water to stop the cooking process. Drain again and run through a salad spinner to remove any excess water. Chop the greens coarsely and set aside.
Step 3Heat a large saute pan over medium-high heat until hot. Add 2 tablespoons olive oil, then the onion, and cook, stirring frequently, until the onion is softened and translucent, 6 to 8 minutes. Stir in the garlic and cook until aromatic, about 1 minute. Stir in the chorizo and cook until it is lightly browned, about 3 minutes. Add the white wine and simmer, stirring to scrape any flavoring from the bottom of the pan, until the wine is almost completely evaporated or absorbed. Remove from heat and spoon the filling into a bowl.
Step 4In the same pan, heat the remaining 2 tablespoons olive oil over medium-high heat. Add the potato and cook, stirring frequently, until it just begins to brown. Stir in 1/2 cup broth. Cover the pan and steam the potato until the pieces are crisp-tender, about 5 minutes, stirring occasionally. Stir in the greens and remaining broth and cook, stirring frequently, until the broth and any liquid from the greens are almost completely absorbed, 2 to 4 minutes. Remove from heat. Stir the potatoes and greens in with the chorizo mixture. Stir in the cheese. Cover the bowl and refrigerate the filling until it is chilled, about 1 hour.
Step 5Divide the prepared pie dough into 12 even pieces, about 3 ounces each, and shape each into a small disk. On a lightly floured board, carefully roll each piece into a circle about 6 inches in diameter and about one-eighth-inch thick. The dough will be flaky and will probably crack on the edges; without working the dough too much, gently mold the dough with your hands as it's rolled to form as perfect a circle as possible. Carefully set the circle aside and continue rolling until all of the circles are formed.
Step 6Assemble the hand pies: Brush the inside of each circle with a very light coating of beaten egg, brushing all the way to the edge of the circle. Place roughly one-fourth cup of the filling in the center of each circle, slightly off to one side (to make it easier to fold over the dough to form the hand pie), but leaving a 1-inch border around the edge on one side. Carefully — this can be tricky — fold over half of the dough, lining up the edges to form a half-circle; you may need to support the dough as it's folded over to keep it from cracking. Gently press down the edges to seal the pie. Trim the edges to clean them up, or gently brush the top of the edge of each pie, then fold the edge in again for a cleaner-looking side. Continue until all of the pies are formed.
Step 7Place the prepared pies on a parchment-lined baking sheet and refrigerate them, uncovered, for 30 minutes. Meanwhile, heat the oven to 400 degrees.
Step 8Remove the chilled pies and brush them with the egg wash. Use a small knife to slash 2 to 3 small steam vents in the top of each pie.
Step 9Bake the pies on the center rack until the pastry is puffed and golden brown, about 30 minutes. Rotate the pies halfway through baking for even coloring. Cool on a rack. The pies can be served warm or at room temperature.