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Preheat oven to 350F.
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In the bowl of a stand mixer fitted with a paddle attachment, beat butter, oil, brown sugar, and granulated sugar until light and fluffy. Add eggs, one at a time, and beat until combined. Beat in vanilla.
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In a separate mixing bowl, whisk together flour, baking soda, baking powder, and salt. Slowly add this mixture to the butter mixture and beat until just combined. Beat in white chocolate chips and chopped Cookies and Cream Bars, in short pulses so you don't break all the chips and bar pieces.
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Take 1 tablespoon of dough and roll into a ball. Place on parchment lined baking sheet. Repeat, placing dough about 2 inches apart. Bake for 10 - 12 minutes or until cookies are a light golden brown. Cool on baking sheet for a couple minutes before transferring to a wire rack to cool completely.