Place butter in a saucepan over medium heat. Stir frequently until the butter turns a golden brown and smells nutty (about 7-11 minutes). Remove from heat, transfer to a bowl, and allow it to cool for about 20 minutes.
In a separate bowl, mix together the all-purpose flour, cornstarch, and baking soda. Set aside.
In a stand mixer or large bowl, combine the cooled browned butter with both the brown sugar and granulated sugar. Beat until light and fluffy (about 3-5 minutes).
Add the egg + 1 egg yolk and vanilla bean paste to the creamed butter and sugar mixture. Beat until everything is fully combined and smooth.
Slowly add the dry ingredients to the wet ingredients in two parts, mixing on low just until incorporated.
Gently fold in the white chocolate chips and chopped sandwich cookies. Stir just until combined to avoid overmixing.
Using a kitchen scale or estimate, form cookie dough balls weighing about about 4-5 oz each. Shape the dough balls into taller, cylindrical forms to achieve a thicker, taller cookie. Press extra cookie chunks or chips into the tops of the cookies for extra chocolate goodness.
Place the cookie dough balls on a baking sheet and freeze overnight. This step is essential for achieving the tall shape.
Preheat the oven to 390°F.Place the cookies directly on a baking sheet from the freezer about 2 inches apart. Bake for 13-15 minutes, until the edges are golden brown and the tops are no longer shiny. Oven times may vary, so watch for the cues of browning with no shininess on top.
Let the cookies cool on the pan for at least 1-2 hours to allow them to fully set and maintain their tall shape.