Position racks in the upper and lower thirds of the oven and preheat to 350 degrees F. Line 2 baking sheets with parchment paper. Whisk the flour, baking powder and salt in a medium bowl. Beat the butter and sugar in a large bowl with a mixer on medium-high speed until fluffy, about 3 minutes. Beat in the egg and vanilla. Reduce the mixer speed to medium low; beat in the flour mixture in 2 batches until combined.
Transfer the batter to a resealable plastic bag and snip the corner to create a 1/2-inch opening. Pipe the batter into 1/2-by-3-inch strips, about 2 inches apart, on the prepared baking sheets. Bake, switching the pans halfway through, until the cookies are golden around the edges, about 15 minutes. Let cool 5 minutes on the baking sheets, then transfer to racks to cool completely.
Line a baking sheet with parchment. Put the chocolate in a microwave-safe bowl and microwave in 30-second intervals, stirring, until smooth. Dip the cookies halfway in the chocolate, letting the excess drip off, then place on the prepared baking sheet. Refrigerate until the chocolate just sets, 10 to 15 minutes.