In a large bowl with a handheld electric mixer, beat cream cheese and powdered sugar on medium speed until smooth and creamy. Add vanilla and milk and mix well. Fold in Cool Whip.
Remove 2/3 of the cream cheese mixture and place 1/3 in one bowl and the other 1/3 in another. In the first bowl, add cocoa powdered and whisk until combined. In the second bowl, add the pureed strawberries and stir until combined.
To assemble the the pie, pour the chocolate mixture into the pie crust and smooth out with a rubber spatula. Place the pie in the freezer and allow it to set, about 30 minutes. Once set, pour the plain layer on top of the chocolate and smooth it out. Return to the freezer for 30 minutes. Finally, add the strawberry layer, and smooth it out. Freezer for 6 hours or overnight. Decorate with whipped cream and a dusting of cocoa powder.