To make the cake, first preheat oven to 350F and spray a bundt pan with nonstick baking spray.
In a small bowl, whisk together the ground flaxseed with the water and set aside to thicken.
In the bowl of a stand mixer, combine the flour, baking powder, baking soda, salt, and sugar.
Add in the vegan butter, and beat on low until it's incorporated and has a sandy texture.
In a small bowl, whisk together the oil, flaxseed mixture, nondairy yogurt, vanilla extract, peppermint, and add them to the dry ingredients. Mix until combined.
Finally, pour in the unsweetened nondairy milk, and beat on medium-low until smooth and creamy.
Gently fold in the white chocolate chips and peppermint pieces.
Pour batter into prepared pan and bake for 45-50 minutes, until cake is set in the center and a toothpick comes out clean.
Let cool slightly in the pan, then turn out onto a serving dish to finish cooling.
To make the glaze, combine the powdered sugar with the extracts. Add in the milk, 1 spoonful at a time, until you reach your desired consistency. I promise you will use less milk than you think! Add food coloring as needed to reach your desired color.
Spoon the glaze on top of the cake, and decorate with crushed peppermints. Enjoy!