Lightly beat egg with water and set aside. Heat oven to 400 degrees. Place 4 deep crocks or ramekins (1-1/4-cup capacity) on a baking sheet. Unfold pastry on a lightly floured surface; roll out seams in dough with a rolling pin. Cut dough into 4 pieces; trim each piece to fit over tops of crocks with an additional 1/2-inch on each side. Place dough on a waxed-paper-lined plate and refrigerate.
Heat 1 T. of the oil in a large nonstick skillet over high heat until oil is shimmering. Saute beef, in 2 batches, 1-1/2 minutes per batch, until lightly browned. Transfer to a plate. Add remaining 1 T. oil to skillet with mushrooms and onions; saute over medium-high heat 4 minutes until mushrooms are softened and onions are translucent.
Stir in rosemary, tomato paste, and garlic, then sprinkle flour over mixture; stir for 1 minute. Add stock, bring to a boil, and cook until thickened, about 3 minutes. Stir in peas and carrots, salt, and pepper, then beef cubes.
Spoon mixture into crocks. Brush outsides edges of crocks lightly with egg wash. Place pastry cut-outs over top of crocks and press dough overhang against outside edges of crocks to adhere. Cut vent holes in centers; brush pastry with egg wash. Bake 24 minutes or until pastry is golden and puffed.