Preheat the oven to 350°F (175°C). Line baking sheets with parchment paper or silicone liners.
Whisk together the flour, baking powder, baking soda, and salt. Set aside.
Using an electric mixer on medium speed, beat the butter and sugar until light and fluffy. Add the egg and vanilla, and mix well.
Reduce the mixer speed to low. Add the flour mixture in three portions, alternating with two portions of the buttermilk. Mix just until combined. (The dough will be thick and slightly sticky.)
Drop the dough by 2-tablespoon portions onto the prepared pans, leaving about 3 inches between the cookies.
Bake, one pan at a time, for 15 to 18 minutes, or until the edges are browned and the centers are set.
Cool the cookies on the pans on wire racks for 10 minutes. Then transfer the cookies from the pans directly to a wire rack to cool completely.
Notes
Yield: Makes about 30 cookies.
To store. After the tea cakes have cooled completely, place them in an airtight container and store them at room temperature for up to 3 days.
To freeze. Place the cookies in a single layer or layered between sheets of parchment paper in a freezer-safe bag or container. Properly stored, they should keep in the freezer for up to 3 months. Thaw them at room temperature before serving.