In a large pot simmer water, salt, sugar, and rosemary sprigs until salt and sugar dissolve; remove from heat. Add ice cubes. Add 1 cabbage cut into steaks (reserve the two ends for another use). Chill overnight.
Drain cabbage; pat dry. Brush with 1/4 cup olive oil; season with crushed rosemary. Grill over medium heat 10 minutes, flipping once.
Meanwhile, in a food processor combine parsley, white wine vinegar, shallot, remaining olive oil, and drained capers until smooth. Serve with cabbage.