Preheat the oven to 350°F (180°C). Line two baking sheets with parchment paper and set aside.
In a medium bowl whisk together 2 cups all-purpose flour, 1/2 teaspoon baking powder, 1/4 teaspoon baking soda, and 1/4 teaspoon salt. Set aside.
In a large bowl using a mixer fitted with the paddle attachment, beat 10 tablespoons softened unsalted butter and 1 cup granulated sugar on medium-high speed until smooth and creamy, about 2 minutes.
Add 1 large room-temperature egg and 2 1/2 teaspoons pure vanilla extract to the butter mixture and beat on medium speed until combined, about 1 minute. Scrape down the sides and bottom of the bowl.
Reduce mixer speed to low. Add the dry ingredients to the butter mixture in three additions, stopping twice to add the sour cream: add one-third of the dry ingredients, then add half of the 1/3 cup full-fat sour cream; add a second third of the dry ingredients, then add the remaining sour cream; finally add the last third of the dry ingredients. Beat only until just combined—the dough will be very thick.
Grease a 1/4-cup measuring cup or use a cookie scoop, and drop mounds of dough about 4 inches apart on the prepared baking sheets (about 6 cookies per sheet).
Bake the cookies for 16–18 minutes, or until the edges are lightly browned.
Let the cookies cool on the baking sheets for 5 minutes, then transfer them to a wire rack to cool completely before icing.
While the cookies cool, make the vanilla icing: in a medium bowl whisk together 5 1/2 cups sifted powdered sugar, 6 tablespoons of the whole milk, 2 tablespoons light corn syrup, 1 teaspoon vanilla extract, and 1/2 teaspoon salt until smooth. (If needed, whisk longer to remove lumps.)
Transfer 1 cup of the vanilla icing to a separate bowl. To that separate bowl add the remaining 1 tablespoon of milk and 3 tablespoons unsweetened cocoa powder; whisk until the chocolate icing is smooth and uniform.
Spread the vanilla icing onto half of the cooled cookies. Refrigerate those iced cookies for 15 minutes or until the vanilla icing is set.
After the vanilla icing has set, spread the chocolate icing onto the other half of the cookies. Allow all the iced cookies to sit until the icing is fully set, about 1–2 hours, before serving.