In a small saucepan, combine the diced strawberries, 2 tablespoons granulated sugar, cornstarch, and lemon juice. Simmer over medium heat, stirring, until the mixture thickens and becomes glossy, about 5–7 minutes. Remove from heat and cool completely.

Beat the softened cream cheese and 1/3 cup sugar until smooth. Mix in the egg yolk, 1 teaspoon vanilla extract, and 1 tablespoon flour until fully incorporated. Chill in the fridge for 10 minutes to firm slightly.

Keep the puff pastry cold. Unfold each sheet and cut into 6 rectangles (12 total). Work quickly to prevent warming.

Spoon about 1 tablespoon of the cheesecake filling into the center of each rectangle, then top with a small spoonful of the cooled strawberry filling. Fold the dough over to form a turnover, sealing and crimping the edges with a fork.

Brush each turnover with the egg wash and sprinkle coarse sugar if you like extra crunch. Bake at 400°F (200°C) for 15–18 minutes, until puffed and golden.

Whisk the powdered sugar with 1–2 tablespoons milk or cream and 1/2 teaspoon vanilla until smooth. Drizzle over warm turnovers and let set slightly before serving.