Add salt, sugar, and butter to the bowl of a stand mixer fitted with a paddle attachment. Beat until light and fluffy for about 3 minutes. Stop and scrape the sides of the bowl with a silicone spatula as needed.
In a small bowl, add egg and vanilla. Using a fork, mix together. With the mixer on medium-low speed, add the egg mixture into the butter and sugar. Beat until well incorporated, scraping the sides of the bowl with a silicone spatula as needed.
In two servings, add flour to the bowl, mixing on low speed until the ingredients combine, scraping the sides of the bowl with a silicone spatula between the flour additions.
Gather dough onto a surface, divide into two portions, shape each into a disk, cover with plastic wrap, and chill for 1 hour.