In a food processor or blender, combine 1 1/4 cups rinsed blackberries; 1/4 cup each catsup, honey, firmly packed brown sugar, and minced fresh ginger; 1 teaspoon pepper; and 1/2 teaspoon salt (optional). Whirl until berries are puréed. Add 1 to 2 teaspoons hot sauce to taste. Pour into a 1-quart glass measure (or combine ingredients in glass measure and mash with a mixer or potato masher). Cover loosely with microwave-safe plastic wrap, leaving vents for steam. Cook in a microwave oven at full power (100%), stirring occasionally, until berry mixture is reduced to 1 1/3 cups, about 8 minutes.