Combine the flour, salt and pepper in a large ziplock bag. Toss the beef, a few pieces at a time in the flour mixture. Set aside.
Heat 2 tablespoons vegetable oil in a large skillet over medium-high heat until shimmering. Add the chunks of beef, in small batches, and cook until the beef is lightly browned on all sides. (Add 1 more tablespoon oil and heat until hot if needed between batches) Remove the beef with a slotted spoon to a large dutch oven or soup pot.
Deglaze the skillet by adding 1/4 cup of beef broth and stir, scraping up any remaining bits of the beef and flour. Add the liquid to the dutch oven.
Add the remaining 1 or 2 tablespoons vegetable oil to the skillet and heat until shimmering. Add the onion, mushrooms and carrots and season with salt and pepper. Sauté for 5 minutes then add the minced garlic. Cook and sauté for another minute then pour the vegetables into the dutch oven.
Add the tomato paste, thyme, stout and remaining beef broth to the dutch oven. Heat over medium-high heat until the mixture begins to boil. Reduce the heat to medium-low and gently simmer the stew with the lid slightly askew, stirring occasionally, until the beef is fork tender, about 90 minutes.
Add the frozen peas and check the seasoning. Remove the stew from the heat.
Preheat oven to 400°F. Prepare a rimmed baking sheet with 4 individual (2-cup) oven-proof bowls, mugs, ramekins or mini casserole dishes. Using a slotted spoon, scoop the stew mixture into each of the bowls. Add enough of the remaining gravy to not quite cover the filling.
Place the sheet (or sheets if making more than 4 bowls) of puff pastry on a lightly floured, clean work surface. Cut the pastry into 4 equal pieces. Working with one at a time, roll the pastry to a size about 1-inch larger than the bowl. Using your finger, dampen the rim of the bowl with a little of the egg yolk/water mixture. Drape the pastry over the rim and trim leaving at least a ½-inch overhang. Crimp, press and seal the pastry to the rim of the bowl. Using a small sharp knife, cut a few slashes in the top of the pastry to vent. Brush the tops with the egg yolk. Decorate if desired with the scrap puff pastry. Repeat with the remaining stew and pastry.
Bake for 20-25 minutes or until the tops are golden brown. Serve and enjoy!