Add the flour and peppermint extract, if using, to the Basic Recipe dough. Divide into quarters. With the food coloring, tint one-quarter of the dough light pink and another quarter darker pink, leaving two quarters of the dough white. Wrap each quarter separately in plastic wrap and refrigerate until lightly chilled, about 30 minutes.
Roll out each quarter of dough between pieces of plastic wrap to about a 7-inch square (1/4-inch thick). Stack the squares starting with the white, then light pink, then white, then dark pink. Cut the stack into 2 small rectangles. Stack one rectangle on top of the other and lightly press them together. Cut the stacked dough into 6 slices (light slices of bread). Lay the slices side by side on a piece of plastic wrap so the stripes are vertical. Cover with more plastic, and using a rolling pin gently roll the dough up and down until the dough is 1/4-inch thick. Still in the plastic, transfer the dough to a baking sheet and refrigerate for 30 minutes.
Preheat the oven to 350°F. Line 2 baking sheets with parchment paper. Remove the top layer of plastic wrap. Align a 5-inch cookie cutter diagonally and cut out as many candy canes as possible. Transfer the shapes to the prepared baking sheets, and bake until slightly golden, about 9 to 11 minutes. Transfer to a wire rack and cool completely.
In a medium bowl combine the flour, baking soda, and salt.
In a separate large bowl beat the butter and sugars with an electric mixer until light and fluffy, about 2 minutes. Add the eggs, one at a time, beating well after each addition. Beat in the vanilla. Gradually add the flour mixture, beating until just combined.
Divide the dough in half, and proceed with a cookie recipe. (Each recipe uses half of the Basic Recipe.) To save the dough for later use, shape each half into a disc, wrap in plastic, and freeze for up to a month.