In a food processor, pulse together flour, salt, and 2 teaspoons sugar. Add the cold cubed butter and pulse just until the mixture looks like coarse meal, about 10 seconds. With the motor running, slowly pour in 1/4 cup ice water through the feed tube. Add more ice water, 1 tablespoon at a time, if needed, until the dough starts to clump together. Do not process more than 30 seconds total.

Divide the dough into two equal pieces. Wrap each in plastic wrap, flatten into a disc, and refrigerate for at least 1 hour. The dough can be kept chilled for up to 2 days or frozen for up to 1 month.

In a medium saucepan, combine the 2 tablespoons butter, diced peaches, 3 tablespoons sugar, and a pinch of salt. Cook over medium-high heat, stirring often, until the butter melts and the mixture bubbles. Reduce the heat to a simmer and cook for 15 to 20 minutes, stirring occasionally, until the peaches are soft.

In a small bowl, whisk together arrowroot starch and 2 teaspoons water. Stir this slurry into the peach mixture and simmer for 1 to 2 minutes, until thickened. Transfer to a bowl and chill until ready to use.

Line two baking sheets with parchment paper. On a lightly floured surface, roll one disc of dough to about 1/8-inch thickness. Cut into 4 1/2-inch squares, re-rolling scraps once. Place the squares on a prepared baking sheet and refrigerate for 15 minutes. Repeat with the second disc. Preheat the oven to 400°F.

Working with one or two squares at a time, place 2 teaspoons of the chilled peach filling in the center of each square. Top with one or two thin peach slices and a sprinkle of sugar. Lightly brush the edges of the pastry with water, then fold the square diagonally to form a triangle. Press the edges with a fork to seal, and cut a small vent on top with a sharp knife. Place the assembled turnovers on the baking sheets and refrigerate for 10 minutes.

Just before baking, brush the tops of the turnovers with heavy cream and sprinkle with the remaining 1 tablespoon sugar. Bake for 26 to 30 minutes, until golden brown. Let cool on the baking sheets for 5 minutes, then transfer to a wire rack to cool completely.

Store leftover turnovers overnight at room temperature. To reheat, place them in a 300°F oven until warm.