Generously flour work surface then roll dough into a rectangle about 9 inches wide and 18 inches long, reaching under dough every so often to make sure it's not sticking. From the long sides, fold the dough like a business letter, bringing one third over the middle third, then the remaining third over that.
From one short end, roll the dough into a tight package, then press down with your hand to make a rough square. Wrap well in plastic wrap and let chill for at least 2 hours before using.
(You can keep the dough refrigerated for 2 to 3 days. If you freeze it, defrost it overnight in the refrigerator.)
To make filling: Cut cherries in half, then combine with granulated sugar, cornstarch, pinch of salt and lemon juice in a medium saucepan. Cook over medium heat, stirring often, until filling thickens, about 10 minutes. Set aside to cool.
Preheat oven to 400 degrees. Prepare a baking sheet with a piece of parchment paper, or leave ungreased.
To make turnovers: Place dough on a well-floured work surface and roll into rectangle a little larger than 12 inches by 16 inches. With a ruler, mark dough into 4-inch increments, then with a knife or pizza cutter, cut into 12 (4-inch) squares, making a clean cut along the rough edges.