1. Preheat the oven to 350°. Spread the pistachios on a rimmed baking sheet and toast for about 7 minutes, or until fragrant. Let cool.
2. Line 2 large baking sheets with parchment paper and place them in the freezer. In a heavy medium saucepan, combine the cream with the sugar, corn syrup and butter and bring to a boil. Stir just until the sugar dissolves. Cook over moderately low heat, without stirring, until the caramel reaches 240° to 244° on a candy thermometer; this can take up to 45 minutes.
3. Fill the sink with 1 inch of cold water. Add the 3 tablespoons of water to the caramel and boil over high heat for 5 minutes, stirring occasionally. Briefly dip the saucepan in the cold water to stop the cooking, then let the caramel cool to 200°.
4. Stir the pistachios and dried cherries into the caramel. Spoon scant tablespoons of the caramel onto the chilled baking sheets and let cool completely, at least I hour.
5. In a medium bowl, melt three-quarters of the chocolate in a microwave. Stir in the remaining chocolate until melted. Dip the bottom of each caramel into the chocolate and return to the baking sheets. Refrigerate for 10 minutes to set the chocolate.
Make Ahead: The caramels can be stored at room temperature in a cool, dry place for up to 1 month.