Make the frosting: Add 1 1/4 cups powdered sugar and 2 teaspoons vanilla extract to the bowl of cream cheese. Beat with the whisk attachment on medium speed until smooth, 3 to 4 minutes. Reduce the mixer speed to low and gradually pour in 3 cups cold heavy cream. Continue beating until the mixture thickens and is smooth, 4 minutes. Increase the speed to medium and beat until the frosting is light, fluffy, and doubled in volume, 1 to 2 minutes. Transfer 1 cup of the frosting to a small zip-top bag and set aside.