Roll out the dough to between 1/4 and 1/8 inch thick. Fit into a deep-dish pie pan, fold the edges under and crimp the edges. Use a fork to lightly prick the bottom of the crust all over (this is called docking the crust, and will help prevent it from puffing up too much during the pre-bake). Freeze the crust in the pan for 30 minutes while you preheat the oven.
Preheat the oven to 400 degrees F. Lightly brush the crust all over with egg white (you’ll have an white leftover since you’ll only use an egg yolk in the dough recipe). Lay a piece of parchment paper over the crust and fill with dried beans or pie weights.
Bake the crust at 400 for 20 minutes. Carefully, remove the paper and the weights, and bake, uncovered, for an additional 8 minutes. Remove from the oven and reduce the oven temperature to 325.
Meanwhile, while the crust is baking, you can prepare the filling.
In a small saucepan over medium-low heat, melt the butter. Continue to cook the butter, swirling the pan occasionally, until the butter is fragrant and nutty brown solids have formed at the bottom of the pan. Pour the butter, scraping up the brown solids, into a large mixing bowl.
Whisk in the brown sugar and corn syrup until smooth. Whisk in the eggs, vanilla, salt and flour until well combined. Mix in the chopped pecans.
Set the pre-baked crust on a baking sheet. Mix up the filling again and pour into the crust. Arrange the pecan halves on top of the filling (overlapping them slightly so that they all fit vertically) to resemble a Christmas tree.
Very carefully, so as not to slosh the filling, return the pie to the oven. Bake at 325 for 50-55 minutes; when done, the filling should still jiggle ever so slightly, and will finish cooking and setting up as it cools. You can cover the edge of the crust with a pie shield or strips of foil partway through baking, if it is getting too brown.
Cool the pie completely on a wire rack before cutting. This pie stays fresh-tasting for days and days after baking.