In a shallow bowl or deep plate, stir together the flour, salt, and pepper. Pat the beef dry with a paper towel. Coat the beef on all sides in the flour mixture and then shake off the excess flour. Transfer the floured beef to a separate plate.
In a large Dutch oven or oven-proof pot, heat the oil over medium-high heat on your stove top until very hot. Brown the beef in batches, adding only enough to cover the bottom of the pan each time. Turn regularly to brown beef on all sides. This should take about 5 minutes for each batch of beef. Transfer the browned beef to a clean bowl or plate and set aside.
Into the same large pot (without cleaning), add the onion and garlic and cook, scraping up any brown bits from bottom of pot and stirring frequently, until the onion is softened, about 5 minutes. If necessary, you can add a bit of water to the pot if it seems too dry and things are sticking too much. Add the tomato paste and cook, stirring, for 1 minute. Stir in the browned beef along with any juices accumulated in bowl. Add the broth, beer, Worcestershire sauce and thyme and bring this to a simmer over medium heat. Cover the pot and transfer to the pre-heated 350° oven. Note: If you’re pot doesn’t have a lid, just cover it tightly with tin foil, instead.
Allow meat to cook in the oven until the beef is very tender and sauce has thickened, about 1 1/2 hours. If the sauce is not thick enough at this point, cook a little longer with the lid off, to cook off some of the liquid. Remove from oven and let partially cool before proceeding. The stew needs to cool a bit before assembling pies, or it will melt the uncooked pastry topping.
Place baking dish or dishes onto a shallow baking pan and pre-heat oven to 425° F.
If making individual pies, divide the cooled stew among bowls. They won’t be completely full. Alternately, spoon all the meat mixture into a baking dish. With a lightly floured rolling pin, roll out the puff pastry dough into a square about 1/8 inch thick. Trim the edges to make an even square, if necessary. For four individual pies, cut the dough into four equal pieces. For one pie, leave whole. Stir together the egg and water and brush a 1-inch border of egg wash around edge of pastry. Invert pastry over each bowl and drape it over the sides, pressing lightly to help it stick to the sides of the bowls. Brush the tops of the pastry with some of remaining egg wash.
Bake pies in the preheated 425° oven until the pastry is puffed and golden brown, about 20 minutes. Reduce the oven temperature to 400° and bake 5 minutes more to fully cook the dough through.