Add ½ cup + 2 tablespoons strawberry puree to a small saucepan.
Heat mixture over medium-low heat, and simmer, stirring occasionally until reduced by about 2/3 to 3 tablespoons, about 10 - 14 minutes (measure and if it isn't quite 3 tablespoons 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, and then freeze until cool (stir occasionally for even chilling, it will chill quickly in the freezer since it's such a small amount).
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 if not using the paddle attachment that constantly scrapes bowl).
Mix in 1 cup powdered sugar, and then blend in 3 tablespoons reduced strawberry puree, vanilla extract, and optional red food coloring.
Add remaining 1 ½ cups powdered sugar and continue to mix until frosting is pale and fluffy, adding in an additional ¼ cup powdered sugar if needed to reach desired consistency.