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Place the white chocolate in a small, microwave-safe bowl and heat on 50% power in 15-second intervals, stirring after each interval, until chocolate is melted and smooth. Set aside to cool for 5 minutes before proceeding.
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Place the cream cheese in a medium mixing bowl and beat with an electric mixer set to medium until cream cheese is light and fluffy, 2 to 3 minutes. Add the powdered sugar and beat until very creamy, stopping to scrape the sides and bottom of the bowl as necessary.
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Beat in the heavy cream, vanilla, and the cooled chocolate. Turn mixer to high and beat for 2 minutes.
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Use a rubber spatula to fold in the chocolate chips and about three-quarters of the cookie dough. (Reserve the remaining cookie dough to use as garnish on the finished cake—store it in the freezer until you’re ready to use it.)
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Scrape the filling into the frozen crust. Smooth the filling into an even layer, then cover with plastic wrap and freeze overnight.