In the bowl of a stand mixer, beat your room temperature butter, shortening and cream cheese until light in colour and creamy. This will take a few minutes on medium-high speed. Scrape down the bowl well (there may be some non-creamed butter stuck at the bottom, so make sure you catch this now and re-beat to avoid clumps at the end).
Gradually add your powdered sugar about 1 cup at a time. *When mixing in powdered sugar, start your mixer on low speed until mostly incorporated, then whip on high speed until smooth.
Scrape down the bowl and then repeat step 2 until you've reached 4 cups of powdered sugar. Now is a good time to taste your frosting to see if it's sweet enough, but keep in mind the remainder of the ingredients will alter the sweetness. If it's too buttery, add more powdered sugar, about a 1/2 cup at a time until you reach your desired sweetness. (I ended up using about 4 1/2 cups).
In a blender, blend your raspberries until they become a uniform liquid. Be sure to give it a shake or a stir to make sure that there are no pieces left behind. I like to also blend for an extra minute or two at the end to chop up the raspberry seeds into smaller pieces since I don't normally strain them out. However, if you plan on straining out the seeds (it's completely up to you) you can skip these extra minutes.
Pour (or strain) the blended raspberries and vanilla extract into the frosting and mix on medium speed until incorporated. Your frosting should be a nice pink colour. Don't forget to scrape down your bowl to fully incorporate all ingredients throughout.
Taste test your frosting again to see if it needs a bit more sugar or even raspberries. Feel free to adjust as needed.
Using a spatula, mix your finished frosting by hand to try to squish out some air bubbles. This should take a few minutes, but it's worth it, especially if you're planning to pipe. If you left in the seeds and are piping, I would recommend using a bigger piping tip so that the seeds don't end up plugging the tip.
Frost your cake or cupcakes (make sure the cakes are cooled before frosting so that they don't melt the frosting!) and enjoy!