In a stand mixer fitted with a paddle attachment, or using a hand mixer and a mixing bowl, beat the butter, powdered sugar, salt, and vanilla extract until smooth and pale.
Gradually add the beaten egg, a little at a time, beating well after each addition.
Add the flour and mix on low just until no dry ingredients remain and a soft dough forms.
Divide the dough into six equal portions and flatten each into a disc. Wrap in plastic wrap and refrigerate until firm but still pliable (will take about 10 minutes in the freezer). The dough can be refrigerated for up to 4 days or frozen for up to 2 months.
Roll out one piece of dough at a time between two sheets of parchment paper to ⅛ inch (3 mm) thick. Place it into the tart pans and trim any excess.
Prick the tart bottoms with a fork and freeze the tart shells while the oven preheats to 375F (190C).
Bake for 20–25 minutes, or until the tartlets are golden brown. Remove from the oven and allow them to cool completely before assembling.