In the bowl of an electric mixer fitted with the paddle attachment on medium speed, or by hand, cream together the butter, pudding mix, and granulated sugar until light and fluffy.
With the mixer on low speed, add egg and milk and mix until incorporated. In a large bowl, whisk flour, baking powder, and salt. With the mixer still on low, slowly add the flour mixture until just combined. Do not overmix!
Divide the dough in half. On a lightly floured surface, roll each half into a log, making a 6" long roll. Place each half on a piece of parchment or waxed paper, and securely wrap. Refrigerate the dough until firm, about 3 hours.
Preheat oven to 375 degrees. With a small, sharp knife, cut the logs into 1/2″ thick slices. Place the slices 1″ apart on ungreased sheet pans and bake for 8 to 10 minutes, until lightly golden. Cool on pan for 2 minutes then transfer to a wire rack to completely cool.
In a small bowl, whisk together powdered sugar and 2 teaspoons lemon juice, adding more lemon juice as needed to reach a drizzling consistency. Drizzle over cooled cookies. Let stand until icing is set.