Heat oven to 375°F. Line 2 baking sheets with parchment paper. In a large bowl, whisk together the flour, baking powder, and salt. In another large bowl, beat the butter and sugar with an electric mixer on medium-high speed until light and fluffy, 2 to 3 minutes. Beat in the eggs, milk, and vanilla. Reduce the speed to low and gradually add the flour mixture, mixing just until incorporated.
Using a ¼-cup measure, place scoops of the dough 2-inches apart on the prepared baking sheets. Bake, rotating the sheets halfway through, until the edges begin to brown, about 15-17 minutes. Transfer to a wire rack to cool completely.
Make the icing: in a large bowl, combine the powdered sugar, ⅓ cup boiling water, and lemon juice. Whisk until the mixture is smooth and spreadable. Divide the mixture in half and portion into two small heat proof bowls. Set one bowl set over a saucepan of gently simmering water. Add the chocolate and corn syrup and stir until the chocolate has melted. Lower the heat and let stand over the hot water. (If the white frosting begins to stiffen up, set it over the hot water for a few moments and stir gently.)