Prep: Preheat the oven to 350F. Spray or line a large baking tray with parchment paper or a silicone mat and set aside. Prepare the flax “egg” by mixing the flaxseed and water well; set aside for at least 5 minutes to let thicken.
Cream the Butter: Add the softened butter, cane sugar, and brown sugar to a large bowl or stand-mixer. Use a stand mixer or electric mixer to cream the butter and sugar together on low speed for 3 to 5 minutes, until the butter is smooth, light, and fluffy. Scrape the sides of the bowl with a spatula as necessary.
Wet Ingredients: Add the flax “egg” and vanilla to the bowl and beat again, until the mixture is smooth and combined.
Dry Ingredients: Add the flour, cornstarch, baking soda, baking powder, and salt. Mix well until a thick, non-sticky dough forms. Add the chopped cookies and white chocolate chips (if using) to the batter and fold into the dough, using a spatula to break up the cookies slightly.
Form the Cookies: Use a cookie scoop or spoon to form 12 cookies with just under 3 tablespoons of dough per cookie. Roll each cookie into a smooth ball and space evenly across the prepared baking sheet. Do not flatten the cookie dough; it will spread as it bakes in the oven.
Bake: Bake in the middle rack of the oven for 10 to 12 minutes; the cookies will look puffy and pale, but the bottom edges should be golden.
Cool: Take the cookies out of the oven and let them rest on the baking sheet for another 10 minutes, where they will deflate further. Transfer to a wire cooling rack and let cool completely.
Serve & Store: Enjoy the cookies immediately, or store in an airtight container for up to 5 days. You can also store the baked cookies in the freezer for up to 2 months, but I would recommend storing the cookie dough balls instead and baking when desired :)