For the filling: In a large Dutch oven (or similar heavy pot with a lid), heat the olive oil over medium heat. Season the beef stew meat with salt & pepper, then add it to the skillet. Brown the meat on all sides. 

Then add the chopped onion, diced carrots, and diced potatoes to the pot. Sauté the vegetables for a 5 minutes, until they just begin to soften.

Add the cremini mushrooms to the pot and cook for another 3 minutes. 

Stir in the minced garlic and tomato paste, and cook for 1 minute, until fragrant.

Slowly pour in the stout beer while stirring constantly. This will help deglaze the pot and create a flavorful gravy. Let simmer for 10 minutes.

Add the paprika, Italian seasoning and white pepper to the pot and give everything a good stir. Let simmer for another 10 minutes.

Pour in the beef broth and stir well. Allow the mixture to come to a simmer, then cover and cook for 1 hour and 30 minutes on the stovetop.

Remove the lid and stir in the peas. Let simmer 10 minutes to warm the peas and thicken the mixture slightly.

Mix the flour with a few spoons of water and add it to the meat and vegetables. Stir vigorously to combine and thicken. Taste test and add more salt & pepper, if desired.

Remove the pot from the stovetop. Lightly spray a 9" x 13" baking dish (or similar-sized oval dish) with nonstick spray, then transfer the hot beef stew filling to it. Let the filling cool to room temperature. Meanwhile, prepare the pie crust.

For the pie crust: Preheat the oven to 350° F.

In a medium bowl, whisk together the flour and salt. 

Add the cold butter cubes to the flour mixture. Use a pastry cutter, fork, or your fingers to incorporate butter into the flour until the mixture resembles coarse crumbs.

Drizzle a few tablespoons of ice water over the crumbs, adding more if needed, until you can gather the mixture altogether with your hands to form a mound of dough.

Roll out the pie crust on a floured surface, to a size and shape that fits the baking dish. and place it over the cooled filling. You can either cover the entire pie with the crust or create a lattice pattern. 

Trim away any excess pie crust, then crimp the edges to seal the pie. Make a few small slits in the top crust to allow steam to escape during baking.

Mix the egg yolk and milk together, then brush the surface of the pie crust with the egg wash.

Bake for 25-30 minutes, or until the crust is golden brown and the filling is bubbling.

Remove from the oven and let the pie cool slightly before serving.