Baking the galette on a pizza stone ensures a very crisp bottom crust. But a baking pan or cookie sheet will work just fine too. Patch any little tears in the crust with extra pieces of dough and smooth them with a wet finger so the juices won’t leak out of the galette. Don’t fret when you are folding over the dough; if the edges are a little rough, they will only add to the rustic character of this delicious dessert.
For the crust
For the apple filling
For the pie crust, whisk the flour, salt, and sugar in a mixing bowl. Work the butter and shortening into the flour using a pastry blender, 2 butter knives, or your fingertips until it resembles coarse cornmeal. Sprinkle in 5 tablespoons ice water and toss together lightly until the dough comes together (add a little more ice water if needed). Shape the dough into a flat disk; don’t overhandle. Wrap the dough in plastic wrap and chill for at least 1 hour.
For the filling, peel and core the apples, then cut them into thick wedges. Put the apples in a large bowl and toss with the sugar and cinnamon. Melt the butter in a large skillet over medium heat, add the vanilla bean, then arrange the apple wedges in a single layer and cook, turning occasionally with a fork, until the apples are tender, about 45 minutes. Scrape the seeds from the vanilla pod into pan and spoon the pan juices over the apples. Remove from the heat.
Preheat the oven to 375°. Roll out the dough on a lightly floured surface into a 14-inch round. Roll the dough around the rolling pin and unfurl on a cold pizza stone or a large baking sheet. Starting in the middle, arrange the apples on the dough in a tight circular pattern to about 3 inches from the edge. Fold the edge of the dough over the apples, pleating the dough as you go. Brush the dough with the heavy cream.
Bake until the crust is golden, about 45 minutes. Remove from the oven and allow to cool for 10 minutes before slicing.