In the bowl of a stand mixer fitted with a paddle, beat the butter on low speed until creamy, about 1 minute. Add the sugar and salt and beat again on low speed until creamy and combined, 2 minutes. Scrape down the sides and add the egg, egg yolk, and vanilla and beat on low speed until combined. Add the flour and mix on low speed until the dough comes together.
Press the dough into the bottom and up the sides of a 10 in [25 cm] tart pan with a removable bottom (see note, page 104).Press and smooth the dough with your hands to an even thickness. Place the pan on a sheet pan and transfer to the freezer until the dough is firm, 20 to 30 minutes.
Adjust an oven rack to the middle position and preheat the oven to 350°F (180°C).
Remove the pan from the freezer and line the pan with parchment paper, covering the edges to prevent burning. Fill the center with pie weights.
Bake for 24 to 28 minutes, until the dough is light golden brown and no longer wet. Remove the tart pan from the oven and carefully remove the pie weights and parchment paper. Brush the center of the tart with the egg wash. Return the pan to the oven and bake for 3 to 6 minutes, until deep golden brown. Transfer the pan to a wire rack and let cool completely.