Preheat oven to 350°F. Line two baking sheets with parchment paper.
Make the flax egg by mixing ground flaxseed and water in a small bowl. Let sit for 5 minutes until thickened.
In a large bowl, cream together the vegan butter, cane sugar, and brown sugar until smooth.
Add the flax egg, vanilla extract, and plant milk. Mix until fully combined, scraping down the sides as needed.
Add the flour, cornstarch, cream of tartar, baking soda, and salt. Mix just until a soft dough forms.
In a small bowl, stir together the sugar and cinnamon for the coating.
Scoop dough using a cookie scooper. Roll into balls, then roll generously in cinnamon sugar. Place on prepared baking sheets, spacing cookies about 2 inches apart.
Bake for 11–13 minutes, until the edges are set and lightly golden. Let cool on the baking sheet for at least 10 minutes before serving.
Optional: Sprinkle with extra cinnamon sugar while hot.