Step 1Heat the oven to 400 degrees. Defrost the puff pastry sheet according to package directions. Butter a large rimmed baking sheet or jellyroll pan.
Step 2Mix the berries with one-fourth cup sugar, the tapioca and cornstarch. Let them sit for 15 minutes.
Step 3Roll out the puff pastry to an 18- by 12-inch rectangle on a lightly floured board. Brush the edges with some of the egg white. Spoon the berries lengthwise down the center of the pastry.
Step 4Pull one side of the pastry sheet up and over the top of the berries and tuck it under the berries inside the other part of the pastry sheet. Pull the other side of the pastry sheet up and over and tuck it under the pastry roll. Press the ends down to seal and tuck under. Brush the entire surface with the remaining egg white. Place in the freezer for 10 minutes to chill.
Step 5Bake until golden brown, about 30 minutes.
Step 6Remove the strudel from the oven and cool slightly. Combine the remaining 1 1/2 tablespoons sugar with the vanilla extract. Sprinkle over the surface of the strudel. Serve warm or room temperature.