Preheat oven to 375. Whisk the flours, corn starch, baking soda, baking powder and salt in a bowl.
In a stand mixer fitted with the paddle attachment, cream the butter until light and fluffy. If your butter is still kind of cold, just whip it for a while until it warms up, or if you have a metal bowl and a kitchen torch you can warm it up that way, which I do often.
Add both sugars and the vanilla and keep whipping until it is light and fluffy again.
Add the egg and yolk and blend until well combined, scraping down the bowl.
Put the mixer on low and add the dry ingredients and once they are starting to blend, add in the cookies and white chocolate chips, along with the strawberry puree if adding. Line 2 baking sheets with parchment.
Use a cookie scoop to scoop out balls of dough. You can use whatever size you like, just adjust the baking time accordingly. Use clean, slightly damp hands to shape into round balls and bake for 7-8 minutes depending on your oven and the size of the cookies. The tops should be puffed up and cracking slightly but they should still look a little under done inside the cracks. Overbaked cookies are the worst, so go on the side of under baked for the first batch and let them cool before baking another try to see if they need more or less time.
Cool on a cooling rack.