Step 1

Preheat the oven to 400 degrees F (200 degrees C). Position oven racks in top and bottom thirds of oven. Line two large rimmed baking sheets with parchment paper.

Step 2

Beat butter, vanilla, baking powder, and salt together in the bowl of a stand mixer fitted with a paddle attachment on medium-high speed until creamy, about 3 minutes, scraping down the sides of the bowl as needed. Beat in sugar on medium-high speed until smooth, about 1 minute. Add egg; beat on low until just combined, scraping down the sides of the bowl as needed. Alternately add flour and milk on low speed, beginning and ending with flour, until just combined.

Step 3

Drop slightly rounded tablespoons of dough 2 inches apart onto the prepared baking sheets. Roll dough with floured hands into balls, then gently flatten each ball to a 1 1/2-inch wide circle.

Step 4

Bake in the preheated oven, rotating pans halfway through, until golden brown on bottoms, about 10 minutes. Place pans on wire racks; let cool completely on pans, about 30 minutes.

Step 5

Meanwhile, bring 1 inch of water in a small saucepan to a boil over medium-high heat; reduce heat to medium. Place chocolate chips, cream, corn syrup, and vanilla for icing in a medium, heat-proof bowl; place bowl over simmering water in the saucepan; cook over medium heat, whisking constantly, until chocolate is melted and mixture is combined, about 5 minutes. Remove bowl from heat; whisk in 1 cup plus 2 tablespoons confectioners' sugar and salt until smooth and thickened, 2 minutes.

Step 6

Dip the flat side of each cookie into the warm icing; return to the baking sheet, chocolate-side up. Let cookies and chocolate cool slightly, uncovered, to room temperature, about 10 minutes. Spread remaining icing evenly over chocolate-dipped cookie tops (about 1 tablespoon each). Let sit at room temperature, until icing is completely set, about 1 1/2 to 2 hours.