In a small bowl, whisk together the flour, baking soda, and salt. Measure out the buttermilk and stir in the vanilla. Set both aside.
In the bowl of an electric mixer, cream the butter and the sugar until fluffy, about 3 minutes. Add the egg and beat another minute more. Mix in the flour mixture in two batches, alternating with the buttermilk.
On a parchment paper lined baking tray, drop 1/4 cup of the batter (about a standard sized ice cream scoop), two inches apart.
Bake for about 14-16 minutes. Transfer to a wire rack to cool completely.
For the Icing:
Make the icing by combining sugar, agave nectar, lemon juice, vanilla, and 1 tablespoon of water in a small bowl. Whisk until smooth.
Pour half of the icing in another bowl and mix in the cocoa powder.
Add the other tablespoon of water to thin the chocolate icing until its consistency matches the white icing.
Frost the cookies by spreading the white icing on one half of the flat side of the cookie, and the chocolate on the other.
Notes
Adapted from Black-and-White Cookie recipes from Epicurious and The New York Times.