In a large frying pan over medium heat, add the olive oil.
Add the onions to the hot oil and cook, while stirring often, until tender and starting to brown on the edges, about 2-3 minutes.
Add the garlic to the onions and cook, while stirring, until fragrant, about 1 minute.
Add the ground beef to the onion mixture and cook, while crumbling the beef, until the meat is no longer pink, about 5-7 minutes.
Add the salt and the black pepper to the beef mixture and stir to combine.
Sprinkle the flour over the beef mixture and cook, while stirring, until the raw flour smell disappears, about 1 minute.
Add the beef stock and the Worcestershire sauce to the beef mixture and stir to combine.
Bring the beef mixture to a boil, while scraping the bottom of the pan to release the browned bits.
Reduce the heat to medium-low and allow the beef mixture to simmer until it has reduced and thickened, about 6-8 minutes.
Pour the beef mixture into a bowl and allow it to cool to room temperature.
Cover and refrigerate the beef mixture until completely chilled, about 1-2 hours.
Line a baking sheet with parchment paper.
Transfer the puff pastry sheets from refrigerator and roll the sheets to an 1/8-inch thickness.
Cut the puff pastry sheets into 12 (4-inch) rounds and 12 (3-inch) rounds. You may want to pre-mark the rounds before cutting to ensure you have enough of the puff pastry sheets mapped out.
Transfer the puff pastry rounds to the prepared baking sheet.
Let the puff pastry rounds chill in the refrigerator for 30 minutes.
Lightly grease a 12-cup muffin tin.
Place 1 of the large puff pastry rounds into the base of each prepared muffin cup, pressing along the sides so they are flush with the bottom and the sides of the tin. The dough should slightly overlap the rims of the muffin cups, so stretch the dough a little as needed.
Transfer the beef mixture from the refrigerator.
Add the white cheddar to the beef mixture and stir to combine.
Evenly divide the beef mixture between pastry rounds in the muffin cups, packing each cup to the top.
In a bowl, add the egg and beat it.
Using a pastry brush, brush the edges of the pastry overlapped in the muffin cups with the beaten egg.
Top each of the beef mixtures in the cups with 1 of the smaller puff pastry rounds, pressing around the edges with a fork to create a seal.
Cut a small slit into the top of each of the pies.
Brush the tops of the pies with the beaten egg.
Freeze the pies for 15 minutes.
Preheat the oven to 425 degrees F.
Bake the pies until golden-brown, about 20-25 minutes.
Allow the pies to cool for 10 minutes.
Serve.