Preheat the oven to 375˚F.
Stir together flax meal and water, and let it sit to thicken for 15 minutes.
Melt vegan butter in a small saucepan over low heat on the stove. Add it to a stand mixer with both sugars and beat until it forms a thick paste.
In a separate bowl, whisk together gluten-free flour, oat flour, cocoa powder, baking soda, and salt.
Add the flax "egg," peppermint extract, and vanilla extract to the butter mixture and mix until combined.
Gradually add the dry mixture to the wet ingredients, folding in both types of chocolate.
Use a large cookie dough scoop to arrange dough on a parchment-paper-lined baking sheet. Freeze for 30 minutes.
Bake 5-6 cookies at a time for 12 minutes. (9-10 minutes for smaller cookies).
Optional: After baking, use a fork to press the edges of the cookie for crinkly, crispy edges.
Let cookies cool completely. Melt dark chocolate and crush candy canes. Dip each cookie in chocolate and top with crushed candy canes. Let chocolate set, then enjoy!