In a medium bowl, whisk together all of the dry ingredients: cake flour, all-purpose flour, cornstarch, baking powder, baking soda, and salt.
In a stand mixer using the paddle attachment, mix the cold butter and brown sugar on medium speed until completely smooth, about 2-4 minutes.
Scrape down the sides and bottom of the bowl in the stand mixer. Add egg, rehydrated ube, and ube extract to the bowl. Mix until combined, about 10-20 seconds.
Scrape down the sides and bottom of the bowl again. Add all of the dry ingredients to the stand mixer and pulse a few times until it is mostly incorporated. Add all of the white chocolate and shredded coconut. Pulse a few more times.
Divide the dough into 5 pieces, about 5-6oz each. Roughly shape the dough into tall mounds using your hands—do not compact them, roll them into balls, flatten the dough, or use an ice cream scoop. Place them on a lined baking sheet or plate, and freeze for at least 30 minutes. Freezing ensures that they do not spread out while baking.Note: You can store the frozen dough in an airtight conainer in the freezer for up to 3 months. You can bake them right from the freezer, but you will have to increase the baking time by 2-3 minutes.
Place oven rack in the middle of the oven. Preheat the oven to 400°F/204°C.
Line a baking sheet with parchment paper or silicone mat. Place each dough mound about 3 inches apart on one baking sheet (you will only be able to fit about 4-5 cookies on one sheet!). Bake for 10-13 minutes. They are done when the tops are a little brown.
Remove baked cookies from oven. Allow them to cool completely on the baking sheet for about 20-30 minutes and do not touch them (they will continue cooking on the sheet)! They are very soft/delicate right after baking and need to set.