Cured Duck with Braised Red Cabbage and Asparagus

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1 whole duck, backbone and wing joints removed, cut in half through breast plate

4 cups sugar

4 cups salt

1 tablespoon garlic, minced

1 tablespoon rosemary

1 tablespoon thyme

1 head purple cabbage

3 jalapeno peppers, minced

1/2 cup red wine vinegar

1 tablespoon dry mustard

1 tablespoon black pepper

1 tablespoon salt

1 tablespoon celery salt

1/2 cup water

12 spears asparagus, cut on the bias in 2-inch pieces

1 sprig rosemary