Make the crust: In a large bowl, whisk together flour and 1/4 teaspoon salt. Cut in the cold butter cubes using a pastry cutter or fork until the mixture resembles coarse crumbs with pea-sized pieces. Drizzle in ice water 1 tablespoon at a time, mixing gently until dough just comes together. Form into a disk, wrap in plastic, and chill for 30 minutes.
Preheat oven to 375°F (190°C). Roll chilled dough on a floured surface into a 12-inch circle. Transfer to a 9-inch pie pan, trim edges leaving 1-inch overhang, fold under, and crimp decoratively. Line with parchment paper and fill with pie weights or dried beans. Bake for 20 minutes, remove weights and parchment, then bake another 12-15 minutes until deeply golden. Brush hot crust with beaten egg white if desired. Cool completely.
Make the lemon filling: In a medium saucepan, whisk together 1 cup sugar, cornstarch, and remaining 1/4 teaspoon salt. Gradually whisk in the water until smooth. Cook over medium heat, whisking constantly, until mixture thickens and comes to a boil, about 5-7 minutes. Boil for 1 minute, then remove from heat.
Whisk the egg yolks in a separate bowl. Slowly pour about 1/2 cup of the hot sugar mixture into the yolks while whisking constantly to temper them. Pour the yolk mixture back into the saucepan, whisking constantly. Return to medium heat and cook, whisking constantly, until very thick and glossy, about 2 minutes.
Remove from heat and whisk in lemon zest, lemon juice, and butter pieces until completely smooth. Pour hot filling into the cooled pie crust immediately, spreading evenly. Press plastic wrap directly onto the surface if not topping with meringue right away.
Make the meringue: In a clean bowl, beat egg whites and cream of tartar with an electric mixer on medium speed until foamy. Gradually add remaining 1/2 cup sugar 1 tablespoon at a time, beating on high speed until stiff, glossy peaks form and sugar is completely dissolved, about 5-7 minutes. Beat in vanilla extract.
Spread meringue over hot filling, making sure to seal it completely to the crust edges to prevent shrinking. Create decorative peaks and swirls with the back of a spoon. Use a kitchen torch to toast the meringue until golden brown, or place under a preheated broiler for 1-2 minutes, watching constantly.
Let the pie cool at room temperature for 1 hour, then refrigerate uncovered for at least 4 hours or overnight until filling is completely set. Slice with a sharp knife dipped in hot water, wiping clean between cuts for the neatest slices.