Combine blackberries, yogurt, sugar, and clove in a blender and purée on high speed until smooth, about 30 seconds. 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. 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 degrees F on an instant-read thermometer.
