Cook strawberry filling: In a medium saucepan, combine strawberries, sugar, lemon juice, and vanilla. Simmer until softened, then stir in cornstarch slurry until thickened. Let cool completely.
Mix cream cheese mixture: Beat softened cream cheese, powdered sugar, egg yolk, and vanilla extract until smooth.
Cut and fill puff pastry: Roll out puff pastry sheets. Cut into squares. Place a spoonful of cream cheese filling and cooled strawberry mixture in the center of each square.
Seal edges with fork: Fold pastry over filling to form triangles. Seal edges with a fork.
Brush with egg wash: Beat egg and brush over turnovers. Sprinkle with coarse sugar.
Bake until golden: Place turnovers on parchment-lined baking sheets. Bake at 375°F (190°C) for 18–20 minutes until puffed and golden brown.
Cool and serve: Transfer to a wire cooling rack. Brush lightly with melted butter if desired. Serve warm or at room temperature.