Combine 1 cup fresh or frozen raspberries and 1/4 cup lemon juice in a small saucepan. Bring to boiling over medium-high; reduce heat. Cook until they release their juices and juices thicken, about 3 minutes, using a fork to mash berries. Transfer mixture to a heat-safe bowl; add 4 cups confectioners sugar and whisk until smooth.
Combine 3/4 cup confectioners sugar and 3 tablespoons freeze-dried raspberries in a food processor; pulse until berries are powdery. Sift mixture to remove seeds.