Make the crust: Pulse the flour, pecans, granulated sugar, shortening and salt in a food processor until it looks like fine meal. Add the butter and pulse until it is in pea-size pieces. Add the egg and 1 tablespoon cold water and pulse until the dough just comes together. Turn out onto a piece of plastic wrap; use the plastic to help form the dough into a disk. Wrap tightly and refrigerate until firm, at least 1 hour or overnight. (The dough can be frozen for up to 2 months; thaw at room temperature.)
Roll out the dough into a 12-inch round between 2 sheets of floured parchment paper. Line a rimmed baking sheet with parchment and coat with cooking spray. Gently peel off the top sheet of parchment covering the dough, then invert the dough onto the baking sheet. Peel off the top piece of parchment. Refrigerate until ready to use.
Make the filling: Whisk the pumpkin, sour cream, 2 tablespoons granulated sugar, the egg yolk, flour, vanilla, cinnamon, ginger, cloves and salt in a large bowl until smooth. Peel the apples and thinly slice. Transfer to a separate bowl and toss with the remaining 2 tablespoons granulated sugar and the lemon juice.
Spread the pumpkin mixture over the center of the dough, leaving a 2-inch border. Arrange the apples in concentric circles on top of the pumpkin mixture. Use the parchment to lift the edge of the dough and fold over the filling; pleat as needed. Refrigerate at least 30 minutes before baking.
Set a baking sheet on the bottom rack of the oven; preheat to 400 degrees F. Brush the crust with the beaten egg and sprinkle with turbinado sugar. Dot the filling with the butter, then set the baking sheet with the galette directly on top of the hot baking sheet in the oven. Bake until golden, 25 to 30 minutes.
Whisk the apricot jam and 1 teaspoon water in a small bowl. Remove the galette from the oven, brush the apples with the jam mixture while still hot and sprinkle with chopped pecans. Let cool slightly on the baking sheet before serving.