Heat your oven to 350F. Line a cookie sheet with parchment paper or silpat.
In a mixing bowl, combine the vegan butter and sugar. Beat on medium-high speed until light and fluffy, about 1 minute. Add flax, cornstarch, lemon juice, milk, and extract. Beat for another minute to combine. Scoot the ingredients to one side of the bowl.
In the other side of the bowl add the flour, baking powder, and salt. Stir to combine. Use mixer to incorporate butter mixture into the flour mixture. Beat until combined.
Use a small cookie dough dispenser to make bite-size balls of dough. Roll the dough balls between your hands to make smooth balls. Place balls 1/2" apart on an ungreased cookie sheet. Use the back of a glass to lightly flatten each cookie dough ball. Bake 8 minutes until cookies are slightly golden around edges. Remove from oven and transfer to a wire rack to cool completely.
To make the lemon frosting, combine powdered sugar and coconut oil in a bowl. Stir to combine. Add turmeric and lemon extract and stir again. Finally, add plant-based milk 1 teaspoon at a time until desired consistency is reached. Top each cooled cookie with a swirl of frosting.
Store in an airtight container at room temperature for up to 5 days, in the fridge for up to 10 days, or in the freezer for up to 2 months.