1) Preheat the oven to 400°F. Lightly grease (or line with parchment) two baking sheets.
2) To make the cookies: Beat together the butter, sugar, salt, baking powder, lemon, and vanilla till well combined.
3) Beat in the eggs one at a time, beating well after each addition.
4) Stir in the flour alternately with the milk, beginning and ending with the flour. Do this gently; there's no need to beat.
5) Using a muffin scoop or a 1/4-cup measure, drop the dough onto the prepared baking sheets. With wet fingers, or the wet bottom of a measuring cup, flatten/spread the dough to a 3"-diameter circle. Leave 2" to 2 1/2" between each cookie; they'll spread.
6) Bake the cookies for 10 to 12 minutes, until they're set, and are perhaps a very light golden brown around the edges. If there's no sign of brown, that's OK. Cookies baked for 10 minutes will be quite moist. Cookies baked for 12 minutes will be drier, and more "authentic." Bake for 11 minutes for an in-between cookie.
7) Remove the cookies from the oven, and cool them right on the pan. As they cool, prepare the icing.
8) To make the vanilla icing: Whisk together the confectioners' sugar, corn syrup, and hot water. For a nice vanilla flavor, add the vanilla; for the whitest icing, omit it.
9) Spread the icing over half of each cookie. Place them on a rack to set while you make the chocolate icing.
10) To make the chocolate icing: Combine the confectioners' sugar, corn syrup, vanilla, hot water, and espresso powder. stirring till smooth.
11) In the microwave, or in a pan set over very low heat, melt the chocolate.
12) Add the melted chocolate into the sugar mixture, stirring till well combined.
13) Spread the icing on the uncovered half of each cookie. You'll have a generous amount of icing, so don't be afraid to pile it on.
14) Set the cookies back on the rack, and allow them to rest for about 30 minutes, till the icing is set.
15) For best storage, wrap each cookie individually, in plastic wrap, and store at room temperature.
Yield: 2 dozen large (3 3/4" to 4") cookies.