Step 1

Cut 10 tablespoons of the unsalted butter into 10 pieces and place in the bowl of a stand mixer (or large bowl if using an electric hand mixer). Place 2 large eggs on the counter. Measure out 2 tablespoons of the whole milk. Let the butter, eggs, and milk sit at room temperature until the butter is softened, about 30 minutes.

Step 2

Arrange 2 racks to divide the oven into thirds and heat the oven to 350°F. Line 3 baking sheets (or as many as you have) with parchment paper.

Step 3

Place 1 3/4 cups all-purpose flour, 1/2 teaspoon baking powder, 1/2 teaspoon baking soda, and 1/2 teaspoon kosher salt in a medium bowl, and whisk until combined.

Step 4

Add 1 cup granulated sugar to the bowl of butter. Beat with the paddle attachment on medium-high speed until light and fluffy, about 2 minutes. Add the eggs one at a time, beating until the first egg is fully incorporated before adding the next. Add 1 teaspoon vanilla extract and 1/2 teaspoon lemon extract if using, and beat until combined, about 1 minute. Add the flour mixture, room-temperature whole milk, and 1/4 cup sour cream, and beat until just combined, about 30 seconds. Using a 2-ounce cookie scoop or 1/4-cup measuring cup, portion the dough onto the baking sheets about 3 inches apart, 4 to 5 per baking sheet.

Step 5

Place the first two baking sheets in the oven and bake for 8 minutes. Rotate the baking sheets between racks and from front to back. Bake until light golden-brown on the bottom, about 7 minutes more. Let the cookies cool on baking sheets for 5 minutes. Using a flat spatula, transfer the cookies to a wire rack, placing them upside-down. Let cool completely, 20 to 30 minutes. Meanwhile, bake the remaining cookies.

Step 6

Place the remaining 6 tablespoons unsalted butter and remaining 3 tablespoons whole milk in a small saucepan over medium heat. Heat, stirring occasionally, until the butter is melted, 1 to 2 minutes. Remove the saucepan from the heat. Add 3 cups powdered sugar and whisk until smooth.

Step 7

Using a spoon and small offset spatula to frost one half of the flat side of each cookie with the white icing (1 tablespoon per cookie): Spoon the icing onto the cookie and use the offset spatula to spread it. Whisk more milk into the icing a teaspoon at a time if the icing becomes too thick, or whisk in more powdered sugar a tablespoon at a time if it becomes too thin. Return the cookies to the rack.

Step 8

Add 2 tablespoons Dutch processed or black cocoa powder into the remaining icing. If the icing is too thick, whisk in more milk a teaspoon at a time as needed. Frost the other half of each cookie with the black icing. Return the cookies to the rack and let sit until the icing sets, 1 to 2 hours.