In a medium bowl, whisk together all dry ingredients (except sugar substitute) until well combined, set aside.
In a separate bowl - the bowl of a stand mixer fixed with a paddle attachment, or a large bowl, if using a hand mixer - cream butter and sugar until smooth and fluffy.
Be sure to scrape down the sides of the bowl a few times, to make sure everything is incorporated into the mix.
Add the eggs to the mix, one at a time, beating well between each addition. Add peppermint extract, and mix until well incorporated and smooth.
With mixer turned down to low speed, slowly add dry ingredients mix to the mixing bowl of wet ingredients, and carefully mix until well incorporated and smooth.
On a clean work surface, divide dough into 3 balls - 2 larger balls of about the same size, and one smaller ball (about half the size of the other two)
Add food colouring to the balls - red to one of the larger ones, white to the other (optional), and green to the smallest.
Knead in until colouring is evenly incorporated in each dough ball.
Wrap dough balls - separately - in plastic film, chill for 1 hour. This is an important step for gluten free cookie dough, don’t skip it!
Divide each dough ball into 3 equal pieces, and each of those 3 pieces into 4 pieces. You should have 12 pieces of each colour.
Following your choice of directions in the post, roll the dough balls out into thin logs, cut in half, and roll into candy cane cookies. Place on prepared baking sheets as you finish them.
Cover and chill for 30 minutes.
Preheat oven to 375F, line cookie sheets with parchment paper.
Bake cookies for 10-12 minutes, or until golden brown on the bottom.
Once completely cool store in an airtight container until serving.