Preheat oven to 400°F.
Pulse the applies, 1/2 cup of the sugar, the lemon juice, and salt in a food processor until the apples are coarsely chopped, about 5 pulses. Let the mixture sit for 5 minutes, then drain, reserving any accumulated juice.
Chef note: The 2nd time I made this, I mixed a tablespoon of flour with the sugar before I tossed the apples in it. I also mixed the applesauce right in with the apples. I think the flour helped keep it from being quite so runny--it was more like pie filling when it was cooked.
Unfold one sheet of puff pastry on lightly floured counter and roll into a 10-inch square. Cut into 4 5-inch squares. Mound 1 tablespoon applesauce and 1 tablespoon of the apple mixture in the center of each square.
Brush the edges of each square with juice drained off the apples and fold into triangles. Crump the edges together using a fork to seal. Lay the turnovers on a wire rack and freeze for 15 minutes. Repeat with remaining sheet of pastry and remaining applesauce and apple filling.
Toss the remaining 1/4 cup sugar with the cinnamon. Remove the rack from the freezer, brush the turnovers with the reserved apple juice, then sprinkle with the cinnamon sugar.
Lay the turnovers on two parchment-lined baking sheets and bake until well browned, 20 to 26 minutes, rotating and switching the baking sheets halfway through baking. Transfer the turnovers to a wire rack and let cool slightly before serving.