Position a rack in the center of the oven and preheat to 300 degrees . Line a cookie sheet with parchment paper. Using a food processor, mix the flour, butter, sugar and salt for 20 seconds. Add the cherries and pulse until chopped, 20 seconds more.
Turn out the mixture onto a work surface and gather into a smooth, compact ball. On a lightly floured surface, roll the dough into a 7-by-9-inch rectangle, about 1/4 inch thick; cut into 12 rectangles. Prick each cookie 3 times with a fork.
Using a metal spatula, place the cookies about 1 inch apart on the prepared cookie sheet and refrigerate for 20 minutes. Bake until light golden-brown, 30 to 35 minutes. Let cool for 10 minutes before transferring to a rack to cool completely; sprinkle with sugar.