Preheat the oven to 375 F. Line 2 baking sheets with parchment paper or a silicone baking mat and set aside. In a stand-up mixer, with the paddle attachment, cream together the vegan butter with brown and granulated sugars until thick and creamy, scraping down the sides as necessary.
Add applesauce and vanilla extract and beat to combine. The dough may look curdled at this point.
In a large bowl combine oats, flour, baking soda, cinnamon, and salt. With the mixer on low, add the dry ingredients to the butter/sugar mixer in 3 batches, scraping down the sides of the bowl between each batch. Mix until a uniform dough has formed.
Wrap bowl in plastic wrap and place in the refrigerator to chill for 30 minutes before baking.
Once chilled, scoop out 2 tablespoons of dough at a time and round them between your hands. Place rounded cookies on the prepared baking sheet, about 3 inches apart from one another. Repeat until both cookie sheets are filled up. You will likely have extra dough for a 3rd sheet.
Press each cookie down slightly with the palm of your hand and bake for 9-10 minutes. Remove from the oven and let cool on the baking sheet for 5 minutes before transferring cookies to a wire cooling rack to cool completely. They will appear slightly underdone when you pull them out of the oven but will continue cooking on the baking sheet.
Store in an airtight container at room temperature for up to 5 days. The uncooked dough can be stored in the freezer for up to 2 months.