Once puree is cool, place the cream cheese and butter in the bowl of a stand mixer fitted with the paddle attachment. (You can also use a hand mixer and a large mixing bowl.)
Beat until very smooth, then add 4 cups powdered sugar, 1 tablespoon milk, and salt and mix to combine. Beat in the puree until combined. Scrape down the sides and bottom of the bowl, then turn mixer to medium and beat until very smooth. Add additional powdered sugar if frosting is too thin and additional milk if frosting is too thick.
Set one cooled cake layer on a cake stand or cardboard cake circle. Spread or pipe a layer of frosting on top. Top with a second cake layer, then frost and decorate the cake as desired. Slice and serve.