Preheat oven to 350°F. In a medium saucepan, whisk together 1 1/4 cups sugar, 1/3 cup cornstarch, and salt. Gradually whisk in water until smooth. Cook over medium heat, stirring constantly until mixture boils and thickens.
In a small bowl, whisk egg yolks. Temper them by whisking in a small amount of hot mixture, then pour yolk mixture back into saucepan. Cook for 2 more minutes, stirring constantly. Remove from heat and stir in lemon juice, lemon zest, and butter until smooth. Keep filling hot.
For meringue, beat egg whites with cream of tartar and 1 tablespoon cornstarch until soft peaks form. Gradually add remaining 1/4 cup sugar and vanilla, beating until stiff, glossy peaks form.
Pour hot lemon filling into baked pie crust immediately. Spread meringue over hot filling, making sure it touches the crust edges all around to seal. Create decorative peaks with the back of a spoon.
Bake for 12-15 minutes until meringue is golden brown. Cool at room temperature for 1 hour, then refrigerate for at least 3 hours before slicing.