Preheat your oven to 350°F with the rack in the center, then line 2 baking sheets with parchment paper. In a small bowl, whisk together the cornstarch and warm water. Set aside.
Using a hand mixer or a stand mixer with the paddle attachment on, cream the softened vegan butter and brown sugar together on medium speed until smooth, about 2 minutes. Next, mix in the peanut butter, maple syrup, vanilla, and corn starch mixture.
In another bowl, whisk the oat flour, baking powder, baking soda, and salt together. Slowly add this to the wet ingredients and mix on low until incorporated.
Place cane sugar in a shallow bowl or pie plate. Roll balls of dough (about 1 ½ tablespoons of dough per cookie), coat with sugar, and place 2 inches apart on the baking sheets. If you have a cookie scoop, it comes in handy for this part.
With the tines of a fork, make a crisscross indent on each cookie, leaving the cookies fairly thick. Bake until the cookies are barely golden at the edges but still slightly underdone in the center, about 7-10 minutes depending on your oven.
Remove cookies from the oven and let the cookies rest on the baking sheet for another 5 minutes, then gently transfer to a wire cooling rack to finish cooling.