Rub oil over meat and season with 1/2 tsp. salt and 1/4 tsp. pepper. Bake for 30 to 40 minutes, or until a thermometer registers 120 degrees; set aside. Reserve pan juice, skimming the fat. Turn off oven.
While roast is cooking, melt butter in a large, heavy-bottomed skillet and cook mushrooms over medium heat, stirring frequently, until all the liquid is evaporated. Remove from heat and season with remaining 1/4 tsp. salt and 1/8 tsp. pepper; cool. Pat mushrooms evenly over roast and set aside.
On a lightly floured surface, roll pie dough into a rectangle large enough to completely enclose roast. Carefully place roast, topside down, on pie dough and fold up to cover, brushing edges with egg white to seal completely. Transfer roast, seam-side down, to a shallow baking pan and brush with a mixture of egg yolk and water. Refrigerate for at least 1 hour.
Preheat oven to 400 degrees.
Bake for 30 minutes, then reduce heat to 350 degrees and bake for five to 10 minutes, or until thermometer registers 140 degrees and pastry is golden brown. Let stand for 15 minutes before slicing.
Meanwhile, in a small saucepan, boil reserved pan juices and red wine until mixture is reduced by 1/4. Add arrowroot powder and beef broth. Cook the sauce over moderate heat, stirring, for five minutes or until thickened, but do not boil. Serve alongside beef.