1. Combine blackberries, yogurt, sugar, and clove in a blender and purée on high speed until smooth, about 30 seconds.
2. Pour through a fine mesh strainer into a mixing bowl, pressing on the purée with a ladle to strain out seeds. Stir in lemon zest.
3. Add lemon juice and salt in 1/4 teaspoon increments until flavor tastes balanced between sweet and tart. Chill in an ice bath or refrigerate until yogurt registers at least 40°F on an instant-read thermometer.
4. Churn yogurt in ice cream machine according to manufacturer's instructions.
5. Transfer ice cream to airtight container and chill in freezer for at least 4 to 5 hours before serving.