Heat oven to 375 degrees. Arrange parchment on a baking sheet, set aside.
Saute beef and onion on medium high heat until beef is cooked and onion is translucent, about 6-8 minutes. Drain liquid, then add in tomato paste. Saute for 2-3 minutes until tomato paste starts to caramelize.
Add in potatoes, stock, seasonings and vegetables. Scrape the bottom of the pan to release any browned bits. Bring mixture to a boil, reduce to a simmer and cover. Cook for 5-10 minutes, stirring occasionally, until potatoes are tender. Uncover and cook for another 2-3 minutes until liquid is completely absorbed.
Place filling in a bowl and cover, refrigerate for up to an hour or until mixture is cold throughout.
On a lightly floured work surface, unroll 1 pie crust. Cut each crust into shamrocks with a 3 or 4 inch cutter, be sure you end up with an even number of circles. Repeat with the remaining crusts.
On half the shamrocks, place a tablespoon of the chilled filling in the center of each round, making sure to leave a ¼-1/2 inch border. Brush the border edges with the beaten egg. Place remaining shamrocks on top and seal edges together with a fork.
Arrange pies on baking sheet and brush with remaining egg wash. Top with shamrock and brush with egg wash again. Bake for 20-25 minutes, or until crust is golden brown.