In a large bowl, shift together the flour, cake mix, baking soda, and salt. Set aside.
In a separate large bowl (use a hand mixer or stand mixer with paddle attachment), beat the softened butter, granulated sugar, and brown sugar until smooth. Add the egg and beat until fully incorporated, scraping down the sides of the bowl as needed. Mix in the vanilla extract.
Gradually add the dry ingredients to the butter mixture and mix on low speed just until combined. Gently stir in the chocolate chips, crushed peppermints, and crushed Oreos, mixing until everything is evenly distributed throughout the dough.
Cover the bowl and refrigerate the cookie dough for 2 hours. If chilling for longer than 2 hours, shape the dough into balls first, place them on a plate, cover, and return them to the refrigerator until you're ready to bake.
When ready to bake, preheat the oven to 350°F. Line a large baking sheet with parchment paper or silicone baking mats.
Scoop about 2 Tbsp of chilled dough for each cookie and roll into balls. Arrange the dough on the prepared baking sheets, leaving room between each cookie. Keep any remaining dough refrigerated while the first batch bakes.
Bake for 10–12 minutes, or until the edges are lightly golden. The centers will still look soft and underbaked when the cookies come out of the oven, but they will continue to set as they cool.
For perfectly round cookies, use a large round cookie or biscuit cutter that's slightly bigger than the cookies. As soon as they come out of the oven, place the cutter around one warm cookie and gently move it in a circular motion. This smooths out the edges and gives the cookie a nice, even shape.
Let the cookies rest on the baking sheet for about 4 minutes before transferring them to a wire rack to cool completely.