Preheat oven to 325°F. First make the sugar syrup. Place the sugar and ½ cup water into a small saucepan over high heat. Bring to a rolling boil, stir to dissolve the sugar and boil for one minute. Pour into a bowl and let cool for at least 30 minutes.
To make the cookies, place the sugar, oil, eggs, vanilla and lemon juice into a large bowl and beat with an electric mixer on medium speed until combined, about 30 seconds. Add the milk and mix in. Add the flour, baking powder and salt and mix until combined.
Cover two cookie sheets with parchment paper or bake in batches. With a cookie dough scooper or small ice cream scoop, scoop up batter and place it on the cookie sheets, about 2 to 3 inches in diameter and about two inches apart.
Bake for 15 to 17 minutes, or until the edges of the bottoms start to color. The top of the cookies should remain light. As soon as the cookies are out of the oven, use a spatula to move the cookies onto a wire rack to cool. These cookies are best iced as soon as they are cool.
To ice the cookies:
Ice the white side first. Place the confectioners’ sugar into a medium bowl. Add two tablespoons boiling water, vanilla, and lemon juice and whisk well. If the mixture is too thick to spread, add the simple syrup to it by tablespoons until the consistency is thick, but still spreadable. Use a small metal spatula or knife to scoop up 1 to 2 teaspoons of icing and spread it on half of the flat bottom of each of the cookies.
Use your finger to rub off any extra white glaze from the edges. The icing will thicken as you ice the cookies and you will need to add some sugar syrup to make the icing smoother. If the white icing gets too loose, whisk in a tablespoon of confectioners’ sugar. Let dry for a few minutes before icing the chocolate side.
Melt the chocolate, either over a double boiler or in a microwave oven, for 45 seconds and stir, heat for another 30 seconds and stir, and then heat for 15 seconds more, or until all the chocolate is melted. Whisk in simple syrup, two tablespoons at a time, until the mixture is smooth, shiny, and spreadable. If the chocolate icing gets too thick to spread, whisk in a tablespoon or more of simple syrup to get the consistency you want. Spread the 1 to 2 teaspoons of the chocolate glaze carefully on the other half of the cookie, moving the chocolate icing back and forth across the cookie to create a straight line between the chocolate and vanilla sides. Let the icing dry for 30 minutes. Store the cookies in an airtight container for up to two days or freeze for up to three months.