Preheat the oven to 425F and line a large baking sheet with a silicone baking mat, or parchment paper.
In a medium saucepan, combine the strawberries, sugar, water, balsamic vinegar, and cornstarch. Cook over medium-low heat 4-5 minutes, stirring frequently until bubbly and thickened and the strawberries are slightly cooked down. Remove from heat and let the filling cool for a few minutes.
Roll out one puff pastry sheet on a lightly floured surface. Cut the sheet into 4 equal squares.
Spoon about 2 tablespoons of strawberry filling into the center of each piece of puff pastry.
In a small bowl, whisk the egg with 1 tablespoon of water and brush the edges of each square with the egg wash.
Fold each square over the filling from corner to corner to form a triangle. Using the tines of a fork, seal the edges of the triangle.
Brush each turnover with egg. Using a sharp knife, slice a small vent into the top of each turnover.
Bake in the preheated oven for about 20-22 minutes until golden brown. Let cool on the baking sheet on top of a wire rack until slightly warm.
Combine the powdered sugar and 1 tablespoon of milk until smooth. Stir in vanilla. Stir in additional milk a few drops at a time until you reach a drizzling consistency.
*frozen strawberries may be used instead of fresh. Thaw them first, and skip the water when cooking.