Move the oven rack to the middle and lower-middle positions. Preheat oven to 375 degrees Line 2 baking sheets with silicone baking mats or parchment paper, and then set aside.
Whisk together the flours, salt, and baking powder in a medium bowl, and set aside. In the bowl of a stand mixer, or in a large bowl, beat the butter and sugar on medium speed until fluffy and lightened in color, about 3-4 minutes.
Beat in the eggs until combined.
Add in ½ of the flour mixture and beat until just combined. Pour in all of the milk and mix until just combined. Add in the remaining flour and mix until just combined and no flour streaks remain. Fold in the vanilla and lemon zest.
Use a ¼ cup measuring cup or a ¼ cup scoop to scoop the batter onto the prepared baking sheets, making sure to leave 2-3 inches between each scoop of batter. Bake the cookies for 16-18 minutes, rotating the pans halfway through baking. The edges will just begin to brown and the centers will be firm.
Cool the cookies for 5 minutes on the baking sheets and then transfer them to a wire rack to cool completely. Once the cookies are completely cooled, flip them over so that the flat side is facing up.
MAKE THE VANILLA ICING:
Whisk together the water and honey in a small saucepan. Set the saucepan over high heat, and heat until the mixture comes to a boil. Then, remove the honey mixture from the heat. Place the powdered sugar into a medium bowl and pour half of the honey mixture over it. Whisk until the honey is fully incorporated and smooth, adding more of the honey mixture as needed to reach a spreadable, but not runny, consistency. Whisk in the vanilla extract.
FROST HALF OF EACH COOKIE:
Fill a disposable piping bag with ¼ cup of the icing and snip off the end of the piping bag. Draw a straight line with the icing right in the middle of each cookie. Squeeze any remaining icing from the piping bag into the bowl with the extra icing. Use a small offset icing spatula to spread the vanilla icing over half of each cookie.
MAKE THE CHOCOLATE ICING AND FINISH THE COOKIES:
In a small microwave-safe bowl, add the chopped chocolate and cook on 50% power for 30 seconds. Stir and repeat until the chocolate is melted, and then set the bowl of melted chocolate aside.
Add the melted chocolate to the remaining vanilla icing and stir until smooth and combined. Add in more of the honey mixture as needed to reach a spreadable, but not runny, consistency. Use an offset spatula to spread the chocolate icing over the remaining half of each cookie.
Let the cookies sit until the icing is set, about 1 hour. Serve.