Add the powdered sugar to a large bowl. Add in the corn syrup, lemon juice, vanilla extract and 1 tablespoon water, whisking and stirring until all the sugar lumps dissolve and a white glaze forms. If the glaze is still a bit thick, add an additional tablespoon of water. You want the icing spreadable but not liquidy.
Transfer half of the icing to another bowl and whisk in the cocoa powder. Add the remaining 2 tablespoons of water and stir and whisk until a dark chocolate glaze forms – the same consistency of the white one. To frost the cookies, I like to start with the white glaze and use a spoon, drizzling a line down the center. I then use the back of the spoon to frost the remaining cookie. I frost all the white halves first, then do the black. I then let the cookies set for an hour or two so the glaze sets. You can pop them in the fridge or freezer before adding ice cream and the glaze will hold up a bit more.
I suggest serving these as an ice cream sandwich “bar” – by having people scoop their own ice cream in between the cookies and pressing them together. You can make a few ahead of time and store them in the freezer for quick grabbing, but one size of the black and white frosting will get a little gooey and stick to the tray or plate they are set on.