In a large bowl or the bowl of a stand mixer, beat 1/2 cup butter and 1/2 cup cream cheese on medium speed with a hand mixer or whisk attachment until smooth.
Next, add 1 1/4 cups granulated sugar and mix on a medium-high speed for a couple of minutes, until the mixture becomes lighter in color.
Add in 1 egg, 2 tsp peppermint extract, and 2 tsp vanilla bean paste or vanilla extract. Mix on a medium speed until incorporated.
In a separate bowl, whisk together 3 cups flour (use a scale or fluff the flour in the bag, then spoon it into the measuring cup and level it), 1 Tbsp cornstarch, 1 tsp fine salt, and 1/2 tsp baking powder.
Mix the dry ingredients into the butter mixture on a low speed until it is JUST incorporated. Scrape the sides of the bowl with a rubber spatula between additions. At this point, the dough should still be pretty thick and sticky.
Spread out a long piece of plastic wrap and scoop half of the dough onto it. Flatten the dough into a rectangle that's about 1/4 inch thick, then wrap the dough tightly and place it in the fridge to chill.
Color the remaining dough red with a generous squirt of red gel food coloring (if you use liquid food coloring, use 1 Tbsp). Mix until the dough is just evenly colored. Flatten the dough into a rectangle that's about 1/4 inch thick on a piece of plastic wrap. Wrap the dough tightly and place it in the fridge to chill.
Chill both portions of dough in the fridge for an hour or overnight. Preheat the oven to 350°F/175°C right before you start shaping the cookies, and line two light-colored, large baking sheets with silicone baking mats or parchment paper. It will take a bit of time to shape all the cookies, and that should give your oven enough time to fully preheat. Set aside.
Unwrap each portion of cookie dough so that it's sitting on top of the plastic wrap. Use a 1/2 teaspoon measure to scoop a portion of red and white dough, then use your hands to roll the dough out into thin strands that are about 4 inches long.
Twist the two strands of dough together to look like a candy cane. Then gently roll the strips of dough together to smooth them out and move the dough into the shape of a candy cane. This can take a little bit of practice, so don't get discouraged if your first few are a little wonky! It gets easier the more you make! The cookies don't spread much as they bake, so you can place them about 1 inch apart on the prepared baking sheets. I was able to fit about 30 per sheet.
Repeat with the remaining dough, making each cookie one at a time to prevent the dough from drying out. Chill the cut-out cookies (pan and all) one more time in the fridge for an hour or the freezer for 15 minutes to help them keep their shape as they bake.
Bake one sheet of cookies at a time for 12-14 minutes on the middle rack of your oven (bake time can vary based on the cookies' size and the dough's thickness). Keep a close eye on the cookies and be sure to pull them out before the edges start to brown.
Let the baked cookies cool on the pan for about 15 minutes, then transfer to a wire rack to finish cooling and enjoy!