Preheat the oven to 400ºF (if you have convection, turn it on). Line two baking sheets with parchment paper and set aside.
In the bowl of a stand mixer with paddle attachment, add the softened cream cheese and mix on medium speed until smooth.
4 ounces cream cheese
To the cream cheese, add the cold butter, brown sugar, and granulated sugar. Turn the mixer on low for a few seconds, then gradually increase speed to medium-high, mixing for about 3-4 minutes until well incorporated and fluffy.
12 tablespoons butter, ½ cup granulated sugar, ¾ cup brown sugar
Turn the mixer to low speed and add egg and egg yolk then vanilla. Mix on medium for about 30 seconds until well incorporated, stopping to scrape down the sides and bottom with a rubber spatula, as needed.
1 large egg, 1 large egg yolk, 1 ½ tablespoons vanilla extract
With the mixer off, add in both flours, salt, baking powder, and baking soda. Turn the mixer on low speed and stir until just combined. Do not over mix.
1 ½ cups all-purpose flour, 1 cup cake flour, 1 teaspoon baking soda, 1 teaspoon baking powder, 1 teaspoon salt
With the mixer off, add in the white chocolate, semisweet chocolate, and Oreo pieces. Stir on low speed for about 20-30 seconds, just until the mix-ins are incorporated. Stirring too much will crush the Oreos and turn your cookie dough gray. If desired, turn the mixer off and knead the dough with your hands a bit to ensure the mix-ins are evenly incorporated.
4 ounces white chocolate baking bar, 4 ounces semisweet chocolate baking bar, 1 ½ cups (12 oreos) oreos
Portion dough into 2-ounce size balls and place about 2 inches apart on prepared baking sheets. I like to use a large #16 cookie scoop for this. Top each cookie dough ball with a piece of Oreo, if desired.
Bake the cookies in the preheated oven, one tray at a time, for 4-5 minutes if using convection (7-8 minutes in conventional oven) then rotate baking trays and bake for another 4-5 minutes for convection (7-8 minutes for conventional). As soon as the cookies are removed from the oven, use a 3-4-inch round cookie cutter to smooth the edges of the hot cookies, if desired. Place the cookie cutter around each cookie and circle it around, allowing the cookie to bounce off the sides of the cookie cutter.
Allow to cool for 10-15 minutes on the trays then transfer to wire racks to cool completely. Leaving them on the hot trays allows the cookies to continue baking and set more in the middle.