In a large mixing bowl, beat together the cake mix, melted shortening, cake flour, and egg until combined. Add 1 tablespoon of water at a time, mixing after each addition, until a dough forms.
Place the cookie dough onto a large piece of plastic wrap. Flatten into a disc shape, wrap completely in plastic wrap, and chill for 2 hours in the fridge.
Preheat your oven to 350 degrees F. Line a baking sheet with a silicone baking mat or parchment paper. Set aside.
Cut the dough disc into quarters. On a lightly floured surface, roll out 1/4 of the dough until it's 1/16-inch thick. Use a 1.5-inch round cookie cutter (or spice jar lid of similar diameter) to cut out the cookies.
Arrange the cookies on the prepared baking sheet. Bake for 8-10 minutes.
Remove from the oven and let cool on the pan for 1 minute before transferring to a wire rack to cool completely. Repeat with the remaining cookie dough.
Coating
Add the chocolate chips, peppermint extract, and vegetable oil to a large, microwave-safe bowl. Heat at 50% power for 1 1/2 minutes and then stir. Heat for another 30 seconds and stir. Repeat until the chocolate is melted and smooth.
Use a fork to dip one cookie into the chocolate. Hold the cookie over the bowl and let the excess chocolate run off. Place the dipped cookie onto a lined baking sheet (silicone mat, parchment paper, or waxed paper). Repeat with the remaining cookies. Pro Tip: If the chocolate starts to set up, reheat it for 30 seconds at 50% power and stir. You can do this as many times as needed.
Refrigerate the cookies until the chocolate is set and firm.
Notes
If you don't have cake flour you can make your own by mixing 3/4 cup + 2 tablespoons of all-purpose flour and 2 tablespoons of cornstarch.