Will need 3 or 4 cookie sheets covered with parchment or foil; position oven racks in the upper and lower thirds of the oven; preheat oven to 375°.
In a bowl, combine the flour, baking powder, baking soda, and salt; mix well.
In the bowl of a stand electric mixer fitted with paddle attachment, beat the butter and sugar together until combined, then beat in the vanilla.
Add the eggs one at a time, beating smooth after each addition.
Decrease speed and beat in a third of the flour mixture, then half of the buttermilk, and another third of the flour mixture; scrape the bowl and beater often.
Beat in the remaining buttermilk, then the remaining flour mixture; scrape the bowl/beater, then remove the bowl from the mixer and give the dough one final mixing with a large rubber spatula.
Drop tablespoons of the dough 3 or 4 inches apart onto the prepared pans.
Bake cookies for about 15 minutes or until they spread and rise—they should be lightly golden.
Slide the papers off the pans onto racks; after the cookies have cooled, detach them from the paper and store them between layers of parchment or wax paper in a tin or plastic container with tight-fitting lid.