Heat 1 tbsp of butter in 10-inch skillet over medium-high heat; cook mushrooms and onion until tender and liquid is evaporated; season with salt and pepper and cool slightly.
Cut thawed pastry into four portions. Place one-fourth of mushroom mixture in center of each portion of pastry. Spread one-fourth of Liverwurst Spread over one side of each steak. Place steaks, liverwurst side down, over mushrooms; wrap pastry around meat. Trim excess pastry and seal ends. Place pastry-wrapped meat, seam sides down, in greased, shallow baking pan.
Beat egg and water and brush pastry. (You can reroll trimmings to make cutouts. Place cutouts on pastry-wrapped meat. Brush again with egg.) Bake, uncovered in 400 F oven for 15 minutes, until pastry is golden and meat is medium-rare. (To test doneness of meat insert thermometer into pastry; it should register 145°F for medium-rare.)
SAUCE: In medium saucepan, combine broth, wine, shallot and thyme; heat to boiling. Reduce heat; simmer, uncovered, 5-7 minutes (until reduced to 1 1/2 cups). In small bowl, mix 4 tbsp butter and flour until smooth. Add to wine mixture; cook until thickened and bubbly. Simmer 1 minute, stirring constantly. Add parsley. Serve with beef.