1. Pour 1 1/4 cups kosher salt in an 8-inch square baking dish; arrange duck breast halves, skin side up, in a single layer over salt. Top duck with remaining 2 3/4 cups kosher salt, pressing down to pack. Cover and refrigerate 24 hours. Remove duck from salt; discard salt. Rinse duck thoroughly under cold water; drain. Pat duck dry; sprinkle evenly with freshly ground black pepper.
2. Place each breast half on a double layer of cheesecloth. Gather edges of cheesecloth together; tie securely. Hang duck in refrigerator for 2 weeks. Unwrap; cut prosciutto into very thin slices. Store in an airtight container in refrigerator up to 5 days.