Add 1/2 cup + 2 Tbsp strawberry puree to a small saucepan.
Heat mixture over medium-low heat, and simmer, stirring occasionally until reduced by about 2/3 to 3 Tbsp, about 10 - 14 minutes (measure and if it isn't quite 3 Tbsp then return and continue to simmer if it's not reduced to correct amount it will make frosting runny).
Pour reduced puree into a small bowl, then refrigerate until cool
In the bowl of an electric stand mixer fitted with the paddle attachment, whip unsalted and salted butter until very pale and fluffy (nearly white. Stop and scrape down bowl occasionally throughout entire mixing process).
Mix in 1 cup powdered sugar, then blend in 3 Tbsp reduced strawberry puree, vanilla extract, and optional red food coloring.
Add remaining 1 1/2 cups powdered sugar and continue to mix until frosting is pale and fluffy, adding an additional 1/4 cup powdered sugar if needed to reach desired consistency.