Combine the flour, salt, and pepper in a large ziplock bag. Toss the beef in the flour mixture. Set aside.
Heat 2 tablespoons olive oil in a large Dutch oven over medium-high heat. Add the coated pieces of beef, in small batches, and cook until lightly browned on all sides. (Add 1 more tablespoon oil and heat until hot if needed between batches). Remove beef with a slotted spoon to a bowl. Set aside.
Add the remaining 1 or 2 tablespoons olive oil to the pot and heat. Add the shallot and saute for approximately 5 minutes. Add the minced garlic. Saute for another 2 minutes.
Deglaze pot by adding 1/4 cup broth and stir, scraping up any remaining beef bits and flour.
Add the carrots, parsnip, and potatoes. Season with salt and black pepper.
Place the browned beef pieces back into the pot. Stir in the tomato paste, thyme, stout and remaining beef broth. Heat over medium-high heat until the mixture beings to boil.
Reduce heat to low and simmer the stew with the lid slightly askew, stirring occasionally, until the beef is fork tender, about 90 minutes.
While the stew simmers, preheat oven to 400 degrees Fahrenheit.
Unfold 1 pastry sheet on a lightly floured surface. Roll the pastry sheet into a 14-inch square. Press the pastry into a 2 1/2-quart round casserole dish. Trim the excess pastry. Prick the pastry thoroughly with a fork. Place aluminum foil onto the surface of the pastry.
Bake for 25 minutes. Remove the foil. Set aside.
Once the beef is cooked, stir in the frozen peas. Then, using a slotted spoon, scoop the stew mixture into the pre-baked crust in the casserole dish. Add enough of the remaining gravy to not quite cover the filling.
On a lightly covered floured surface roll out the remaining puff pastry sheet to about one inch larger than the diameter of the casserole dish.
Using your finger, dampen the rim of the filled crust with a little of the egg yolk/water mixture. Drape the pastry sheet over the top of the baking dish, leaving at least 1/2 inch overhang. Crimp, press and seal the pastry top crust to the bottom crust. Using a sharp knife, cut a few slashes in the top to vent. Brush the top with the egg yolk mixture.
Bake for 30 minutes until the top is golden brown.
Serve and enjoy!