Preheat oven to 375°F. Line a baking sheet with parchment paper and set aside.
Wash and dry the peaches. Using a paring knife, cut diagonally to remove 1 1/2-inch round piece from the stem end of each peach.
Using heavy-duty kitchen shears, reach into the middle of the peach and firmly grab hold of the pit. Then pull hard to extract it from the peach. Repeat for the remaining peaches (see note).
In a medium bowl, combine oats, flour, sugar and butter. Crumble the mixture together to a coarse texture using your fingertips.
Stuff the mixture into the cavities in the peaches until packed firmly.
Unroll or roll out the pie crust. Using a sharp knife, cut into long strips about 1/2-inch wide (see note).
Wrap the pie dough around each peach, moving from the bottom to the top and pressing the layers gently together with your fingers to seal until completely covered. You'll need to use multiple strips per peach.
Brush the pastry lightly with egg wash and bake for 30-35 minutes until lightly golden.
Remove from oven and cool for 10 minutes. Drizzle with optional caramel sauce before serving.