Preheat your oven to 400°F (200°C). Line a baking sheet with parchment paper.
In a mixing bowl, combine the flour and cold butter. Use a pastry cutter or your fingers to mix until the mixture resembles coarse crumbs.
Gradually add ice water, one tablespoon at a time, until the dough comes together. Form the dough into a disk, wrap in plastic wrap, and refrigerate for at least 30 minutes.
In another bowl, combine the sliced plums, peaches, sugar, cornstarch, and lemon juice. Toss to coat the fruit evenly.
Roll out the chilled dough on a floured surface into a rough circle about 12 inches in diameter. Transfer the dough to the prepared baking sheet.
Arrange the fruit mixture in the center of the dough, leaving a 2-inch border around the edges. Fold the edges of the dough over the fruit, pleating as needed.
Brush the edges of the dough with the beaten egg. Bake for 40 minutes, or until the crust is golden and the fruit is bubbly.
Let the galette cool slightly before serving. Enjoy!