In a medium bowl, use a wooden spoon to combine the flour, baking powder, baking soda, and salt. Set aside.
Place the room-temperature non-dairy butter and granulated sugar in a large bowl. Use a hand mixer to cream the butter and sugar together until smooth and well-combined.
Add the non-dairy milk, lemon juice, non-dairy sour cream, and vanilla extract to the wet ingredients and beat with the hand mixer until completely incorporated.
Add the flour mixture to the wet ingredients and beat to combine, 1-2 minutes. Cover the bowl with plastic wrap and place in the fridge for 1 hour.
Preheat the oven to 350℉. Cover a half sheet pan with parchment paper. Use a medium cookie scoop, or a scant ⅛ cup measure, to portion out the cookie dough onto the sheet pan.
Bake for 18-22 minutes, until the cookies are just golden brown around the edges. Allow them to cool on the sheet pan for 15 minutes, then transfer to a cooling rack to cool completely. When the cookies are cool, turn them over, as you will be icing the (flat) bottoms of the cookies.
Make the icing: combine the powdered sugar with the corn syrup and 3 tablespoons of water and whisk well to combine. Using an icing spatula or small offset spatula, carefully spread the icing onto one half of a cookie. You can use the straight edge of a pastry cutter or parchment paper to make the icing as neat as possible. Repeat with the rest of the cookies. Note that you will have icing left over after icing the vanilla half of the cookies; this remaining icing will become your chocolate icing in the next step.
Allow the vanilla icing to fully set; this may take up to an hour or so. While the icing is setting, add the cocoa powder and another 1-2 tablespoons of water to the remaining icing and whisk well to combine. Once the vanilla icing has set, spread the chocolate icing on the other half of each cookie. Allow the chocolate icing to fully set.
Notes
Store the cookies in an airtight container at room temperature for up to 3-4 days.