First, preheat oven to 400F.
Next, prepare the cheesecake filling. In a medium bowl, combine the cream cheese, ½ cup of sugar, vanilla, lemon extract, and salt. Whisk together until smooth and creamy.
Drain your cherries (don't rinse) and set aside for now.
Lightly flour a clean working surface, and lay one of your puff pastry sheets on it.
Using a floured rolling pin, gently roll out the dough to about 15 in x 15 in - don't worry about being exact! The point is that you want it thinner than it is straight out of the box.
Cut the sheet into 6 even squares, then repeat with the second sheet of pastry.
Spoon about 1-2 tablespoon of the cheesecake mixture on each puff pastry square, making sure to leave a clean border, and top each with 3-4 cherries. It's okay if you don't use all the cream cheese mix - we will turn it into frosting in a minute!
Brush the edges of the pastry squares with water, and seal into triangular shape. Go over the seals by pressing with a fork to secure.
Brush the top of each turnover with heavy cream, and sprinkle on the additional sugar.
Bake for 18-20 minutes, until turnovers are golden brown and crisp.
While they cook, make the frosting! Take whatever leftover cheesecake filling you have, and add in powdered sugar until it thickens into frosting consistency. You will need about a cup of sugar, but add just a bit at a time so you can get the texture you want!
When the turnovers are cooked, let cool to room temperature, then frost and enjoy!