Prepare the baking sheets. Preheat the oven to 325°F (163°C). Then, line two baking sheets with parchment paper and position the oven racks in upper and lower-middle positions.
Make the dough. Cream the butter and brown sugar on medium speed in an electric mixer until light and fluffy, about 2-3 minutes. Then, slowly add 2 cups of flour, mixing on low speed until the mixture appears crumbly. Next, transfer the dough to a floured surface and knead in the remaining ¼ cup of flour by hand. The dough should be slightly crumbly but hold together when pressed. Do not add liquid.
Roll and cut the cookies. Roll the dough to a ½-inch thickness on a lightly floured surface. If the dough sticks, lightly dust the rolling pin with flour. Then, use a round cookie cutter or the rim of a glass to cut out cookies. Re-roll scraps as needed. Place the cookies on the prepared baking sheets, leaving about 1 inch of space between them (they will not spread much).
Bake and cool. Bake the cookies for 18-20 minutes, rotating the baking sheets halfway through to make sure even baking. The cookies are done when the edges are lightly golden. Cool completely on wire racks before decorating.
Crush the candy canes. Place peppermint candies or candy canes in a sealed plastic bag. Use a rolling pin or the bottom of a pan to crush them into small pieces. Set aside.
Melt the chocolate. Combine semisweet chocolate chips and 2 teaspoons of shortening in a microwave-safe bowl. Heat on 50% power for 30-second intervals, stirring between each, until smooth. Alternatively, use a double boiler for melting. Repeat the process with white chocolate chips and the remaining shortening in a separate bowl.
Dip and decorate the cookies. Dip one-half of each cooled cookie into the melted chocolate. Then, use a spoon to help coat evenly if needed. Place the dipped cookies on parchment or wax paper. While the chocolate is still wet, sprinkle generously with crushed peppermint candies. Let the cookies sit at room temperature until the chocolate sets, about 30 minutes. Place them in the refrigerator for 10-15 minutes for faster setting.