Prep baking sheets. Preheat oven to 325 degrees with two racks in the upper middle and lower middle positions. Prepare two baking sheets with parchment paper.
Make dough. Cream butter and sugar in an electric mixer until light and fluffy. Add 2 cups of flour and mix on low until the flour is mixed in. The cookie dough will appear very crumbly, but do not add any liquid. Turn it on onto a floured surface and mix in the remaining ¼ cup of flour by hand.
Roll out and cut. Roll the cookie dough out to ½ inch thickness, dusting the rolling pin with flour if the dough sticks. Cut into rounds using a cookie cutter or biscuit cutter. Place on a prepared baking sheet. They will not spread much.
Bake and cool. Bake the cookies for 20 minutes, until cookies are set and edges are golden brown. Cool completely on wire racks for dipping.
Dip and decorate. Melt the chocolate chips and shortening in the microwave on LOW for 45 seconds, stirring halfway through. Dip the cookies in the chocolate and place them on a piece of waxed or parchment paper. Add the crushed peppermints while the chocolate is still melted. Let them rest on the paper until the chocolate sets.
You can double or triple the dough easily with a stand mixer.