In a large bowl, cream together the softened vegan butter and sugar with a hand mixer (or stand mixer with paddle attachment) until fluffy. Beat in the vegan sour cream, vanilla and almond extract, if using. Scrape down the sides of the bowl as needed.
To the bowl with the butter mixture, add the flour, baking powder, baking soda, cornstarch and salt. Mix on low until just combined.
Shape the dough into a disk, wrap in plastic and place in the refrigerator to chill for about an hour.
Preheat the oven to 375 degrees F.
Roll the dough out on a lightly floured surface, using a rolling pin, about 1/4 inch thick.
Use a 2-3 inch round cookie cutter to cut the cookies and place on parchment or silicone lined baking sheets. Bake for 9-10 minutes until the cookies are very lightly golden on the bottom. Do not over bake or the cookies won't be soft.
Let the cookies cool on the pan for 5 minutes, then carefully transfer to a cooling rack to cool completely before frosting.