Preheat the oven to 350 degrees . Line a 9-by-13-inch baking pan with parchment, allowing a few inches of paper to hang over the two long sides. Press the gingerbread dough evenly into the lined pan.
Bake until the top is dry, 18 to 20 minutes. Transfer to a rack and let cool completely in the pan. Using the parchment paper overhang, lift the cookie out of the pan and invert onto a cutting board; remove parchment.
Place half the frosting in a medium bowl; tint with red food color and spread on the cookie. Divide the remaining frosting between three small bowls. Tint one yellow, one green and leave one white. Place the frosting in three separate resealable bags, snipping off a tiny corner of each. Decorate the cookie with stripes of yellow, green and white frosting, forming a plaid pattern. Cut into 24 squares.