Step 1In a medium saucepan, combine the sugar with the water and bring to a boil. Add the cranberries and reduce the heat to a simmer. Cook until the cranberries have burst and the sauce thickens slightly, stirring frequently, about 10 minutes. Remove from heat and set aside to cool completely (the sauce will thicken more as it cools). This makes about 2 3/4 cups cranberry sauce, more than is needed for the final recipe.
Step 1In a medium bowl, whisk together the yogurt, powdered sugar, cream, lemon zest and juice until smooth. Freeze in an ice cream maker according to the manufacturer's instructions.
Step 2Meanwhile, place the cranberry sauce in a bowl and mash the sauce to break up any large berries using a potato masher, a rubber spatula or the back of a large spoon (don't worry about over-mashing; there will still be small lumps).
Step 3Immediately transfer the frozen yogurt to a medium bowl and gently fold in the cranberry sauce with a large rubber spatula. Do not fully incorporate the sauce; it will look marbleized. Freeze for an additional 30 minutes to 1 hour, until slightly set but still light in texture.
Step 4Serve the frozen yogurt slightly softened, with a small dollop of extra cranberry sauce if desired.