In a 5 1/2 quart Dutch oven over medium high heat, warm 1 Tbsp oil. Cook mushrooms, onions, salt and pepper 12 minutes. Transfer to a bowl. Season beef with salt and pepper. Dredge in flour. In same pot over medium high heat, warm 2 Tbsp oil. Sear one-third of beef 7 minutes; transfer to separate bowl. Add 1/2 cup water to pot; scrape up browned bits. Pour into separate bowl. Repeat twice with rest of beef.
Return pot to medium high heat. Add garlic and tomato paste; stir 30 seconds. Add beef, stout, broth and reserved liquid; scrape up browned bits. Add mushrooms, onions, carrots , potatoes and thyme; bring to a boil. Cover, simmer over medium low heat, stirring occasionally, 3 hours.
In the meanwhile, prepare the pastry:.
In a food processor, combine the flour, salt and sugar and pulse until blended, about 5 pulses. Add the butter and process until the mixture resembles coarse meal, about 10 pulses. Add 1/3 cup of the ice water and pulse 2 or 3 times. The dough should hold together when squeezed with your fingers but should not be sticky. If it is crumbly, add more water 1 Tbs. at a time, pulsing twice after each addition. Turn the dough out onto a lightly floured work surface and shape into a disk. Wrap with plastic wrap and refrigerate for 1 hour.
Remove the dough from the refrigerator and let stand for 5 minutes. Sprinkle the top of the dough lightly with flour, place on a lightly floured sheet of parchment paper and roll out into a 12-by-16-inch rectangle. Sprinkle the cheese over half of the dough, then fold the other half over the cheese. Roll out the dough into a 16 1/2-inch square. Using a paring knife, trim the dough into a 16-inch round.
Refrigerate the dough until firm, about 10 minutes, then lay the dough on top of the beef and stout pie and bake as directed in that recipe. Makes enough dough for a 16-inch round.
Preheat an oven to 400°F.
Brush the rim of the pot with water. Lay the pastry round on top, allowing it to droop onto the filling. Trim the dough, leaving a 1-inch overhang, and crimp to seal. Brush the pastry with the egg mixture, then cut 4 slits in the top of the dough. Bake for 30 minutes. Let the potpie rest for 15 minutes before serving. Serves 8 to 10.