Make the cookies: Position racks in the upper and lower thirds of the oven; preheat to 350˚. Line 2 baking sheets with parchment paper. Whisk the flour, malted milk powder, baking powder, baking soda and salt in a medium bowl.
Beat the butter and granulated sugar in a large bowl with a mixer on medium-high speed until light and fluffy, about 3 minutes. Add the eggs one at a time, beating well after each addition, then beat in the vanilla. Reduce the mixer speed to low and beat in the flour mixture in two additions, alternating with the sour cream, until just combined.
Drop mounds of dough (2 tablespoons each) about 2 inches apart on the prepared pans. Bake, switching the pans halfway through, until the bottoms and edges of the cookies are set and just starting to brown, 16 to 18 minutes. Let cool 5 minutes on the pans, then remove the cookies to racks to cool completely.
Meanwhile, make the icing: Sift the confectioners’ sugar and malted milk powder into a large bowl. Add the corn syrup, vanilla and hot water and whisk until thick but spreadable. (If the mixture is too thick, add more hot water, 1 teaspoon at a time.) Remove half of the icing to a medium bowl. Using a rubber spatula, fold 1/2 teaspoon white food coloring into one bowl of icing, adding more food coloring if needed. Fold 1/2 teaspoon red food coloring and the cocoa powder into the remaining bowl of icing, adding more food coloring if needed.
Flip the cookies flat-side up on the racks. Spread the white icing on half of each cookie using a small offset spatula, sprinkle with white coarse sugar and return to the racks. Refrigerate until set, 15 to 20 minutes. (For a neat line between the 2 colors, lay a piece of foil or wax paper halfway across each cookie before spreading the white icing, then remove after refrigerating.) Spread the red icing on the other half of the cookies and sprinkle with red coarse sugar. Let set at room temperature, at least 2 hours or overnight.