Adjust oven rack to lower third position. Preheat oven to 350 degrees.
Pat beef cubes (2 1/2 pounds) dry of any excess moisture with paper towels. Place cubes in a bowl and generously season cubes with salt and pepper.
Heat 2 tablespoons of vegetable oil in a Dutch oven, over medium-high heat, just until smoking. Brown half the beef in batches, make sure not to overcrowd the pan. Transfer browned meat to the bowl with the raw cubes.
Add remaining 2 tablespoons oil to the Dutch oven and reduce heat to medium. Stir in mushrooms (4 ounces), onion (1), and carrots (2), and saute until vegetables are softened for about 5 minutes. Be sure to stir the bottom of the pan to scrape up any browned bits. Add tomato paste (2 tablespoons) and garlic (1 tablespoon) and cook, stirring constantly, until fragrant, about 30 seconds.
Pour in the wine (1/2 cup) and continue to cook until liquid is evaporated for about 2 minutes. Sprinkle flour (3 tablespoons) over vegetables are stirred to coat. Cook for 1 minute, stirring constantly.
Add broth (1 1/2 cups), Better Than Bouillon and water (1 heaping tablespoon dissolved in 1/4 cup boiling water), soy sauce (1 tablespoon), Worcestershire (1 tablespoon), bay leaf (1), and sugar (1/4 teaspoon) until combined. Add browned and raw beef and bring to simmer. Cover and place in the oven. Cook until beef is tender, about 1 1/4 - 1 1/2 hours.
Take filling out of the oven and increase oven temperature to 400 degrees. Discard bay leaf. Stir in peas (1 cup) and 1 teaspoon chopped thyme. Taste filling and adjust salt and pepper if needed. Let filling cool, uncovered, for 30 minutes, then pour the filling into 9-inch deep-dish pie plate.
Brush the rim of the pie plate and the interior lip of the pie plate with egg (1). Place pie crust over filling so dough overhangs edges of pie plate slightly. Fold overhanging pie dough inward so folded edge is flush with the inner edge of the pie plate. Crimp dough evenly around the edge of the pie using your fingers.
Make a 1/2-inch hole in the center of the pie. Cut six 1/2-inch slits around the hole, halfway between the center and edge of the pie. Brush the crust with the remaining egg, then season with salt, pepper, and remaining 1/2 teaspoon chopped thyme.
Place pie on a rimmed baking sheet and bake until crust is golden brown, about 30 minutes. Transfer pie to cooling rack and let cool for 20 minutes. Serve.