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Pour the flour into a medium mixing bowl and add the butter, lard, and salt. Pinch the fats and mix together with the flour until the mixture resembles fine breadcrumbs.
Mix in the vinegar and 2 tablespoons of water, then gradually add additional water until a smooth pastry forms.
Knead gently a few times, then shape into a disc, wrap tightly in plastic wrap, and chill for at least an hour.
While the pastry is chilling, make the filling. Melt the butter over medium-high heat in a Dutch oven. Once the butter is bubbling, add the onion, celery, carrot, and garlic. Cook for about 5 minutes, until the vegetables have softened.
Stir in the potatoes and cook for another 5 minutes, then mix in the crumbled corned beef until well blended.
Pour in the beef stock, Worcestershire sauce, English mustard, and black pepper and stir until combined. Cover and bring to a boil.
Remove the lid and allow the mixture to simmer for about 15-20 minutes, until the potatoes and vegetables are cooked through, stirring occasionally.
Skim off any liquid that hasn't evaporated, then stir in the fresh chopped parsley.
Pour the filling into a 9 by 13-inch cake pan and chill in the freezer for 20 minutes, stirring halfway through.
Preheat the oven to 400 F. Beat an egg with a fork and set aside.
Roll out the pastry on a lightly floured surface until it's a little less than 1/8 inch thick. Line the base of a 6-cup (1.5 liter) pie dish with the pastry, trimming off any excess.
Evenly spread the cooled filling in the pastry-lined dish, then brush the edge of the pastry with beaten egg.
Re-roll the remaining pastry and cover the pie with it. Press down the edges to seal, then trim off the excess pastry. Crimp the edges as desired, brush the top with beaten egg, and cut a few vent holes in the top.
Bake for 45-55 minutes at 400 F, until the pastry is well browned. Cover the edges with aluminum foil if they brown too quickly.
Let the pie stand for about 15 minutes before serving. Serve hot or cold.