Preheat the oven to 350F and line a baking sheet with parchment paper. Prep the flax egg by mixing together the ground flax and water in a small bowl and setting aside for later.
Add the vegan butter, brown sugar, and white sugar to a bowl. Use a standing/electric mixer to whip together until creamed and fluffy, about 1-2 minutes.
Next, add in the flax egg, molasses, and vanilla and mix together another 30 seconds.
Then, add in the dry ingredients including the flour, rolled oats, baking soda, cinnamon, and salt. Make sure to add the flour and oats first and baking soda and spices on top so everything gets mixed evenly.
Mix everything together until you have a smooth but slightly sticky dough. Then, use a cookie scoop and drop the cookies onto the parchment lined baking sheet. Make sure that each cookie is kept about 2 inches apart since they will spread while baking. You will likely have to add the dough to two separate baking sheets.
Add the tray to the oven and bake for 10-12 minutes until the cookies are fully cooked on the outside with slightly soft centers. Make sure to bake the cookies on the same rack in the oven or in two batches for the best results.
Remove the tray from the oven and gently tap on the counter to help the cookies deflate. Let the cookies cool on the baking sheet for 5 minutes. Then, move to a wire rack to cool completely before adding frosting.