Cut the berries in half and blend to a smooth puree in a blender or food processor. You should have about 1 cup of puree.
Put the puree and lemon juice in a medium sized heavy bottomed saucepan and bring to a boil. Boil, stirring often with a silicone spoonula or spatula to keep the mixture from scorching on the bottom of the pan. Cook the mixture down until it is thick like tomato paste or fruit butter. You want to end up with about 1/4 cup. This will take somewhere around 15 minuteds, give or take. Scrape the thickened puree into a cup and chill. Note: the puree can be made up to a few days ahead if you like.
Cream the soft butter, cream cheese, and cooled strawberry puree (use 1/4 cup) together until smooth and lump free. Note: you can use a stand mixer, or electric beaters. See my method for making frosting in my food processor in the notes below ~ it's genius!
Start adding the sifted sugar, one cup at a time. Scrape down the sides of the bowl often. You should end up with a creamy thick frosting. If it seems too thin, beat in a little more sugar. If it seems too thick, add a bit of lemon juice or milk.
Yeilds enough to frost one layer cake or 24 cupcakes.
You can make this frosting right in a full sized (14 cup) food processor, and there is no need to soften the cream cheese in this case. Put the cream cheese, butter, and puree in the bowl and process until smooth. Add the sugar, a cup at a time, processing between each addition until creamy. Scrape down the bowl as needed.