Beat butter, sugar, eggs, and vanilla extract together in a bowl using an electric mixer until smooth and creamy. Sift flour, cream of tartar, baking soda, and salt together in a separate bowl; mix, 1 cup at a time, into creamed butter mixture until dough is just mixed. Form dough into 1-inch-thick disks and place on a plate; chill in refrigerator, 2 hours to overnight.
Preheat oven to 350 degrees F (175 degrees C).
Roll disks into desired thickness on a lightly floured work surface using a floured rolling pin. Arrange dough on a baking sheet.
Bake in the preheated oven until cookies are just starting to turn golden brown, 7 to 10 minutes.
