Put fresh or frozen cranberries in saucepan and transfer 1/2 cup to a small bowl.
Add 1 cup sugar, 1 strip orange or lemon zest, and 2 tablespoons water to the pan and cook over low heat, stirring occasionally, until the sugar dissolves and the cranberries are soft, about 10 minutes.
Increase the heat to medium and cook until the cranberries burst, about 12 minutes.
Reduce the heat to low and stir in the reserved cranberries.
Add sugar, salt, and pepper to taste and cool to room temperature before serving.