Combine brown sugar, butter, and white sugar in a large bowl; beat with an electric mixer until smooth and creamy. Mix in sour cream, food coloring, and egg.

Combine 2 1/2 cups flour, dark cocoa powder, regular cocoa powder, baking soda, and salt in a separate bowl. Gradually stir flour mixture into butter mixture until well combined. Place dough in the refrigerator to chill for at least 1 hour.

Preheat the oven to 375 degrees F (190 degrees C). Line a baking sheet with a piece of greased and floured parchment paper.

Spread remaining 1/2 cup flour on a clean, dry surface and use some to coat a rolling pin. Roll chilled dough to a thickness of approximately 1/4 inch and cut cookies with whatever shape cookie cutter you like. Place 2 inches apart onto the prepared baking sheet.

Bake in batches in the preheated oven until cookies bounce back immediately after being touched, about 7 minutes. Cool on the baking sheet for 1 minute before removing to a wire rack to cool completely, 8 to 10 minutes more.

Stir 4 cups powdered sugar, Neufchatel cheese, butter, and vanilla extract in a large bowl until creamy. Fold in chocolate chips. Transfer frosting into a resealable plastic bag.

Cut one corner off the plastic bag and pipe frosting onto 1/2 of the cookies. Top with remaining cookies and sprinkle with remaining 1/4 cup powdered sugar.