Make the Homemade Puff Pastry: Add the flour, sugar and salt to a food processor and pulse to combine. Add one-quarter of the butter cubes and process in four 1-second pulses. Add the remaining butter to coat the cubes with flour, two 1-second pulses. Transfer the mixture to a medium bowl.
Combine the ice water and lemon juice in small bowl. Add 3 tablespoons of the liquid to the flour and butter mixture and toss with a rubber spatula until just combined. Toss in the remaining water, then turn the dough out (it will be a shaggy mixture) onto a work surface and press it together with both hands.
Use the heel of one hand against the work surface to drag a small portion of the dough forward in a short, brisk stroke. Repeat with the remaining dough portions. Gather the dough together (using a bench scraper if it's sticking) and repeat the pressing of the dough a second time. Pat the dough into an 8 by 4-inch rectangle, wrap in plastic wrap, and refrigerate at least 30 minutes.
Unwrap the dough and place on a lightly floured large piece of parchment paper and roll into a 15 by 10-inch rectangle. Fold the dough lengthwise into thirds. Starting from a narrow end, loosely roll up the dough. Press it into a 6 by 5-inch rectangle.
Repeat step #4, first chilling the dough for 30 minutes if it is soft and sticky. Once you have repeated step #4, wrap the dough in plastic wrap and refrigerate for at least 30 minutes.
On a large sheet of parchment paper lightly dusted with flour, roll the dough to an 18-inch square Using a pizza cutter or ravioli cutter, trim the dough to a 15-inch square. Cut the dough into nine 5-inch squares. Slide the parchment and dough onto a cookie sheet or large cutting board and refrigerate while preparing the filling.
Grate the apples on the large holes of a box grater. Combine the grated apples, sugar, lemon juice and salt in a medium bowl. Line a baking sheet with parchment paper; set aside.
Working one at a time, remove dough squares from the refrigerator and set on a clean work surface. Place 2 tablespoons of the grated apple filling (squeezed of excess liquid) in the center of the dough square. Moisten two adjoining edges of the dough square with a finger dipped in the apple liquid, then fold the top portion of dough over the bottom, making sure to overlap the bottom portion by ⅛ inch. With a fork dipped in flour, press to seal and crimp the edges. Using a wide metal spatula, transfer the turnover to the prepared baking sheet. Repeat the process with the remaining dough squares. Refrigerate for at least 30 minutes or cover with plastic wrap and refrigerate for up to 24 hours.
When ready to bake, preheat oven to 375 degrees F and adjust oven rack to upper-middle position. Combine the sugar and cinnamon in a small bowl.
Brush the turnovers lightly with water and sprinkle evenly with the cinnamon sugar. Bake until golden brown, 18 to 22 minutes. Using a wide metal spatula, transfer the turnovers to a wire rack to cool completely before serving.