In a small saucepan combine raspberries and lemon juice; bring to a simmer over medium heat and cook, stirring constantly and gently smashing the raspberries, until thickened and reduced, about 10 minutes. Remove from heat.
Using a fine-mesh strainer over a small bowl, pour in the raspberry puree and push through the strainer with a spatual, which will give you smooth puree with no seeds. Set aside to cool.
In the bowl of a stand mixer fitted with the paddle attachment, beat the softened butter on medium-high speed (scraping down the sides as needed) until completely smooth, about 2 minutes. Add in the raspberry puree and beat until completely combined, about 1 minute (the mixture may look very “wet”, but don’t worry!) Add in the powdered sugar, one cup at a time, on low speed, until smooth.
Spread the frosting evenly over the cooled cookie base and add the sprinkles. Cut into squares (I did 4×6 rows, to make 24 bars!) Serve and enjoy!