For the cranberry vodka: Combine the cranberries, sugar, orange peel and 1/4 cup water in a medium saucepan and bring to a boil over high heat. Cook until the cranberries soften and pop, about 5 minutes.
Pour the vodka into a large heatproof jar, then add the cranberry mixture. Let sit at room temperature for at least 3 days or up to 1 week. Strain and chill in the refrigerator before using.
For the cranberry-orange mule: Mix together the ginger beer, cranberry vodka, lime juice and orange slices in a pitcher. Serve over crushed ice and garnish with skewers of Sugared Cranberries.
Combine the granulated sugar and 1 cup water in a saucepan. Bring to a boil and cook until the sugar is dissolved. Add the cranberries and weight with a plate to submerge. Cover the pot and refrigerate for at least 8 hours and up to 48 hours. Strain the cranberries, reserving the syrup. (Refrigerate leftover syrup in an airtight container for up to 1 month.)
Spread the superfine sugar on a baking sheet and toss the cranberries in the sugar. Refrigerate on the baking sheet, uncovered, for 1 hour. Toss again in the sugar before transferring to an airtight container. Refrigerate until ready to serve.