Preheat the oven to 400°F (205°C) and line a large baking sheet with parchment paper. In a large bowl, combine the sliced apples, fresh cranberries, brown sugar, granulated sugar, cornstarch, cinnamon, nutmeg, vanilla extract, and lemon juice. Toss until the fruit is evenly coated, then let it sit for about 10 minutes.
Roll the pie dough into a 12-inch circle on a lightly floured surface and transfer it to the prepared baking sheet. Spoon the fruit mixture into the center, leaving a 2-inch border around the edges, then scatter the butter cubes over the filling. Fold the edges of the dough over the fruit, pleating as you go to create a rustic galette.
Whisk together the egg and milk to make the egg wash, then brush it over the exposed crust. Sprinkle with coarse sugar if desired. Bake for 38 to 42 minutes, or until the crust is deep golden brown and the fruit filling is bubbling.
Let the galette cool on the baking sheet for about 15 minutes before transferring it to a serving plate. Slice and serve warm or at room temperature, with vanilla ice cream or whipped cream if desired.
