In a medium saucepan, combine diced strawberries, sugar, cornstarch, lemon juice, and vanilla extract. Cook over medium heat, stirring constantly, until the mixture thickens and becomes glossy (about 5-7 minutes). Remove from heat and let cool completely.

Preheat your oven to 400°F (200°C). Line a baking sheet with parchment paper. Lightly flour a clean surface and roll out the puff pastry sheet to a 12x12-inch square. Cut the pastry into 4 equal squares (about 6x6 inches each).

Place about 2 tablespoons of cooled strawberry filling in the center of each pastry square. Fold two opposite corners of each square over the filling, overlapping slightly, and press gently to seal. Brush the exposed pastry with egg wash.

Arrange the pastries on the prepared baking sheet, spacing them at least 2 inches apart. Bake for 18-22 minutes, or until the pastry is puffed and golden brown. Transfer to a wire rack to cool for 10 minutes.

In a small bowl, whisk together powdered sugar, milk, and vanilla extract until smooth. Drizzle the icing over the warm danishes. Serve immediately or let the icing set for 5 minutes.