Place cream cheese in a small mixer bowl. With an electric mixer, beat at medium speed until smooth. Beat in sugar and vanilla until well blended. Stir in 1 cup of the whipped topping until blended. Spread in bottom of crust.
Prepare pudding according to package directions, using 1 3/4 cups milk. Pour over cream cheese layer in crust. Refrigerate at least 10 minutes or up to 3 1/2 hours.
Spoon on dollops of remaining whipped topping just before serving.