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Puree the berries in a blender or food processor. Pour into a small saucepan and set the pan over medium heat. Bring to a boil, stirring frequently.
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Turn heat to low and simmer, stirring often, until puree reduces by about half (8 or 9 minutes). You'll have about a 1/2 cup of puree.
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Remove from heat and transfer a scant 1/4 cup of the puree into a small bowl.
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Pour the rest into a second bowl through a fine mesh sieve to remove the seeds. Use a rubber spatula to press the puree through the sieve, extracting as much of the raspberry juices as possible. (This is what you’ll use in the frosting.)
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Set both bowls aside to cool to room temperature.
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Once puree is cool, place the butter in a large mixing bowl and beat with an electric mixer until very smooth.
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Add the seedless puree, 3 1/2 cups powdered sugar, 3 tablespoons milk, and the salt and beat to combine. (Depending on how sweet your berries are, you may want to add a little more salt than this recipe calls for.)
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Scrape down the sides and bottom of the bowl, then turn mixer to medium and beat until smooth. Add additional powdered sugar if frosting is too thin. If frosting is too thick, add additional milk 1 tablespoon at a time to achieve desired consistency.