Preheat the oven to 375 degrees Fahrenheit. Beat the vegan butter, both sugars, and vanilla extract in a stand mixer or in a large bowl using a handheld mixer. Beat on medium speed until smooth (approximately 2 minutes). Pour in the coconut milk, molasses, and pumpkin puree and mix on low speed until combined.
In a separate bowl, whisk together the flour, baking soda, salt, and pumpkin pie spice. Add the flour ingredients to the wet ingredients and mix on low speed until the ingredients are combined.
Refrigerate dough for 1 hour. This step is important because it prevents the cookies from spreading out too much when they cook.
Scoop two tablespoons of dough and form into a ball. Add pure cane sugar to a small bowl and roll the dough in it. Press the ball down slightly.
Arrange on a lined baking sheet, leaving a few inches between each cookie. Bake for 12 minutes, or until the edges are golden. The cookies may seem too soft when you take them out of the oven, but they will firm up as they cool.
After they are finished baking, leave the cookies on the baking sheet for 10 minutes before transferring them to a cooling rack.