Preheat oven to 350°F (180°C). Line 2 cookie 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 unsalted butter (softened) and 1 cup white granulated sugar on medium-high speed until smooth and creamy, about 2 minutes.
Add 1 large egg and 2 1/2 teaspoons pure vanilla extract; beat on medium until combined, about 1 minute. Scrape down the sides and bottom of the bowl.
Reduce mixer speed to low. Add the dry ingredients in three additions (about one-third of the flour mixture each time). After the first and second dry additions, add the 1/3 cup full-fat sour cream in two roughly equal portions (half after the first dry addition and the other half after the second). Beat until the dough is fully combined and very thick.
Grease a 1/4-cup measuring cup or cookie scoop. Drop mounds of dough (1/4 cup each) onto the prepared baking sheets, spacing them about 4 inches apart (about 6 cookies per sheet).
Bake for 16–18 minutes, or until the edges are lightly browned. Remove from oven and let cookies cool on the baking sheets for 5 minutes, then transfer to a wire rack to cool completely before icing.
For the icings: In a medium bowl, whisk together 5 1/2 cups sifted powdered sugar, 6 tablespoons whole milk, 2 tablespoons light corn syrup, 1 teaspoon vanilla extract, and 1/2 teaspoon salt until smooth (this is the vanilla icing).
Transfer 1 cup of the vanilla icing to a separate bowl. Add the remaining milk (the last 1 tablespoon) and 3 tablespoons unsweetened cocoa powder to that bowl and whisk until smooth to make the chocolate icing.
To finish the cookies: Spread the vanilla icing over the entire top of each cooled cookie. Refrigerate the iced cookies for about 15 minutes, or until the vanilla icing is set. Then spread the chocolate icing over half of each cookie to create the black-and-white effect. Return cookies to the rack and allow the icing to set completely, about 1–2 hours, before serving.