Preheat oven to 350 degrees F. Spray a 9-inch pie pan with nonstick cooking spray; set aside.
In a bowl, or stand mixer, combine all crust ingredients until a thick dough forms. Divide dough into two equal parts and roll each half into a circle approximately 11 inches in diameter (you want it larger than the pie pan to form a crust on the edge). Lightly press one half of the dough into the bottom and up the edges of the pie pan (the other half will go on top).
Heat oil in a large skillet over medium-high heat. Add beef and season with salt and pepper. Cook for 2-3 minutes, stirring occasionally, or until just browned on the outside. Use a slotted spoon and remove beef from the skillet and set aside on a plate.
Add carrots, onion and mushrooms to the same skillet and cook for 5 minutes. Stir in tomato paste, garlic and flour and cook for an additional minute.
Stir in beef broth, soy sauce, worcestershire sauce, frozen peas and dried thyme. Add beef back to the skillet and let cook a few minutes or until sauce has thickened.
Pour filling into pie pan and place remaining crust on top then pinch and seal the edges together. Beat egg then brush on top of the crust and cut a few slits in the top of the pie to release steam while cooking.
Bake for 50 minutes or until golden brown on top. Let cool for a few minutes before serving as the inside of the pie will be hot.