Place the raspberries in a small saucepan over medium heat and cook, stirring occasionally, until they release their juices and break down, about 5 to 7 minutes.

Transfer the cooked raspberries to a fine mesh sieve set over a bowl and press them through with a spoon or spatula, scraping the underside to collect the smooth puree; discard the seeds and let the puree cool completely.

Add the room temperature butter to a stand mixer bowl or large mixing bowl and beat on medium speed until pale, creamy, and slightly fluffy, about 2 to 3 minutes.

Turn the mixer to low speed and add the powdered sugar in 3 to 4 additions, mixing until each addition is mostly incorporated before adding the next and scraping down the bowl as needed.

Increase the mixer speed to medium and beat until the mixture is light and fluffy.

With the mixer on low, add the cooled raspberry puree 1 tablespoon at a time, beating well after each addition until the frosting is smooth, thick, and evenly pink.

Beat in the vanilla bean paste and salt just until the frosting is smooth and silky.

Use the frosting immediately for cakes, cupcakes, or cookies, or cover and refrigerate until needed.