Make the crust: Whisk the all-purpose flour, whole-wheat flour, granulated sugar, cinnamon and salt in a large bowl. Work in the butter until it is in pea-size pieces. Mix the sour cream with 3 tablespoons ice water, then stir into the flour mixture with a fork. Turn out the dough onto a piece of plastic wrap, press into a disk and wrap. Refrigerate 2 hours. Roll out the dough on a floured surface into a 12-inch circle. Place on a baking sheet and refrigerate 30 minutes.
Preheat the oven to 400˚ F. Make the filling: Toss the peaches and plums with the granulated sugar, cornstarch and lemon juice. Arrange the fruit over the dough, leaving a 1 1/2-inch border. Fold in the edge of the dough, pleating as needed.
Bake the galette: Brush the edge of the dough with beaten egg and sprinkle with coarse sugar. Bake until golden and bubbling, 45 to 50 minutes.