1. Prepare the flax egg by combining the ground flax and water in a small bowl, then set aside for at least 5 minutes to thicken.
2. Use a food processor knife to chop half of the freeze-dried strawberries into small pieces; set aside.
3. Place the remaining strawberries in a blender or food processor and blend or pulse for 30 to 45 seconds until a fine powder forms; set aside.
4. Add the vegan butter to a large mixing bowl, then add in 1/3 cup of cane sugar and brown sugar. Use a handheld mixer to cream the butter and sugar together over medium speed until light and fluffy, about 3 minutes.
5. Add the thickened flax egg and vanilla to the bowl and beat again until fully incorporated.
6. Add the flour, strawberry powder, baking powder, and salt to the bowl. Beat on the lowest setting until just combined.
7. Add the chopped freeze-dried strawberries to the bowl and fold them into the batter.
8. Cover the cookie dough with plastic wrap and place in the fridge to cool for 60 minutes, or up to overnight.
9. Preheat the oven to 350°F while the cookie dough is chilling and line a baking sheet with parchment paper.
10. Divide the cookie dough into 12 even balls and roll each in the remaining cane sugar. Place on the lined baking sheet and flatten slightly.
11. Bake in the middle rack of the oven for 12 to 13 minutes, or until the edges look set and the centers are soft and puffy.
12. Let the cookies set for 5 to 10 minutes before transferring them to a cooling rack.
13. Enjoy immediately, or store in an airtight container on the counter for up to 4 days.
