In a saucepan over medium heat, stir together the strawberries, sugar and water until the sugar is dissolved. Heat the mixture to boiling, stirring often. Reduce heat and add the vanilla and lemon juice.
In a small cup, combine the cornstarch and two tablespoons of water. Mix well.
Pour this into the boiling strawberry mixture. Stir and boil over medium-low heat for 4 minutes or until the mixture has thickened a bit.
Remove the pan from the heat.
Add 1 tablespoon of butter and mix until the butter has completely melted.
If you'd like your strawberry syrup to have a nice red color, add a few drops of red food coloring. This is completely optional.
Allow the mixture to cool for 15 minutes, then add the last 1/2 cup of quartered, fresh strawberries. Cool completely.
When the strawberry filling has cooled completely, preheat the oven to 400 degrees.
Gently unfold the puff pastry sheets and cut each sheet into 9 squares.
Spoon 2-3 strawberry pieces in the center of each square, draining most of the syrup as you do this. (You can use the leftover strawberry syrup to make a glaze.)
Fold the puff pastry square to make a triangle and then use the tins of a fork to press the pastry closed along the two open edges.
In a small bowl, whisk together the egg and water.
Use a pastry brush to spread the egg wash on top of the puff pastry triangles.
Sprinkle a bit of granulated sugar overtop the egg wash and place the pastries on a baking sheet lined with a silicone baking mat.
Bake for 12-13 minutes until the turnovers are golden brown.
Remove from the oven and allow the turnovers to cool.
If desired, add two tablespoons of the leftover strawberry syrup to 1/2 a cup of powdered sugar and whisk together until smooth.