Add the vegan butter and white sugar to the bowl of a stand mixer and cream them together. Add in the lemon extract and mix in well.
Add the flour, baking soda and salt to a bowl and mix together. Then add the dry ingredients in with the wet ingredients and mix in by hand (don't use the electric mixer for this part). Mix until crumbly.
Add in the lemon zest and mix in.
Then add in the soy milk and mix in very well. It will seem like it is too dry to form a dough, but get in there with your hands until it does. Only add more soy milk if you absolutely have to.
Preheat the oven to 350°F (180°C).
Break off pieces of the big ball of dough and roll them into balls and place onto a parchment lined baking tray. Aim to get 20 balls.
Then flatten the balls either with a fork or by placing a square of parchment paper over a ball and flattening with the base of a glass.
Place into the oven to bake for 12 minutes.
When they are done they will be lightly browned on top but still very soft in the middle. Allow to cool on the tray and they will firm up as they cool.
Prepare your lemon glaze by adding the ingredients to a bowl and mixing in with a spoon.
When the cookies are completely cool, use a teaspoon to drizzle the glaze over the cookies.