How to Make It
Combine 3/4 cup all-purpose flour, whole-wheat flour, granulated sugar, and 1/2 teaspoon salt in the bowl of a food processor; pulse 3 to 4 times to combine. Add butter; pulse 8 to 10 times or until mixture resembles coarse meal. Add milk and 1/4 cup yogurt; pulse 20 times or until pea-size lumps form and dough begins to pull away from the sides of the bowl. Turn dough out onto a work surface; shape into a 6-inch disk. Cover with plastic wrap; refrigerate 2 hours or overnight.
Preheat oven to 425°F.
Combine remaining 2 tablespoons all-purpose flour, 1/4 teaspoon salt, brown sugar, juice, vanilla extract, cinnamon, nutmeg, ginger, and apples in a large bowl; toss to coat.
Unwrap dough; place on a large piece of lightly floured parchment paper. Roll dough to a 15-inch circle. Place parchment paper with dough on a baking sheet. Spoon apple mixture onto dough, leaving a 2-inch border. Fold edges of dough over filling to partially cover. Brush edges of dough with egg white. Bake at 425°F for 25 minutes or until crust is golden brown. Let stand at room temperature 30 minutes. Cut into 8 wedges.
Combine remaining 1/4 teaspoon salt, remaining 1/2 cup yogurt, and vanilla bean paste in a bowl, stirring with a whisk. Serve with galette.