Preheat oven to 350 degrees, with one rack in upper third and another in lower third. In a medium bowl, whisk together flour, baking powder, salt, and baking soda; set aside.
Using an electric mixer on medium-high, beat butter and sugar until light and fluffy, 2 to 4 minutes. Add egg and vanilla; beat well to combine.
With mixer on low, add half the flour mixture, followed by sour cream, then remaining flour mixture; mix just until smooth. (Dough will be stiff; you may need to finish mixing it by hand with a wooden spoon.)
Drop mounds of dough, each equal to about 3 tablespoons, 3 to 4 inches apart, onto two ungreased baking sheets; sprinkle with sugar. Bake until edges of cookies are just firm and tops are barely beginning to brown, 20 to 25 minutes, rotating sheets once halfway through. Transfer cookies to a rack to cool.