PREHEAT oven to 375°. Line 2 baking sheets with parchment paper.
COOKIES: In a large bowl, beat butter and sugars with a mixer at medium-high speed until fluffy, 3 to 4 minutes, stopping to scrape sides of bowl. Add egg, beating well.
WHISK together Swans Down Cake Flour, cocoa and baking soda in a separate bowl. With a mixer on medium speed, gradually add flour mixture to butter mixture, beating until combined. Add food coloring, vinegar and vanilla, beating until combined.
SHAPE cookie dough into balls (about 2 tablespoons each) and place at least 2 inches apart on prepared pans. Refrigerate for 10 minutes.
BAKE until tops are set, about 8 minutes. Let cookies cool on pans for 5 minutes, then transfer to wire racks to cool completely.
FROSTING: In a large bowl, beat cream cheese with a mixer at high speed for 1 minute. Reduce mixer speed to medium and gradually add confectioners’ sugar, beating until smooth.
CRUMBLE 2 of the cooled cookies and set aside. Spread frosting over remaining cookies and sprinkle with cookie crumbles.