Preheat oven to 350 degrees. Generously butter two baking sheets, or line them with parchment paper. Combine the butter and sugar in a mixing bowl; beat until light and fluffy. Beat in sour cream and egg.
In a separate bowl, whisk together the flour, baking powder, and salt. Gradually beat the dry ingredients into the butter mixture until they are well combined. Stir in currants and lemon zest.
Drop heaping teaspoonfuls of dough about 1 1/2 inches apart onto prepared baking sheets. Bake until cookies are puffed and golden around the edges, about 20 minutes. Let cool 5 minutes on baking sheets before transferring cookies to wire racks to cool completely.