Preheat the oven to 350°F and line two baking sheets with parchment paper.
In a large bowl, beat the butter and sugar with a mixer on medium speed for 2-3 minutes, until light and fluffy. Beat in the sour cream, vanilla and lemon extract until smooth.
In a separate bowl, whisk together the flour, cornstarch, baking powder, baking soda and salt. Add the dry ingredients to the butter mixture in three additions, alternating with the milk and mixing on low just until smooth.The batter will be thick and soft, like a very thick cake batter.
Scoop 1/4 cup mounds of batter onto the prepared sheets, spacing them 3 inches apart with no more than 6 per sheet. Use the back of a damp spoon to gently spread each mound into a 3 inch round.
Bake for 14-16 minutes, until the edges are barely golden and the tops spring back when lightly pressed. Cool on the sheet for 5 minutes, then transfer to a wire rack to cool completely.
In a medium bowl, whisk the powdered sugar, 3 tbsp hot water, corn syrup and vanilla until thick, smooth and glossy. Transfer half of the icing to a second bowl and stir in the cocoa powder, melted chocolate and remaining 1 tbsp hot water.Cover the chocolate icing with plastic wrap so it stays soft while you work.
Turn the cooled cookies flat side up. Using an offset spatula, spread the vanilla icing over one half of each cookie and let it set for 15 minutes.
Spread the chocolate icing over the other half, adding a few drops of hot water if it has thickened. Let the icing set for 1 hour until firm. Makes about 12 large cookies.