Preheat a grill for cooking over medium-high heat.
Place the corned beef slices on the grill and cook, flipping halfway through, 4 to 5 minutes. (The slices will contract like bacon and become dry and crisp.) Let cool slightly and tear into bite-sized pieces. (Leave the grill on.) Reserve.
Brush the cabbage wedges generously on all 3 sides with olive oil, and sprinkle with salt and pepper. Grill, covered, cross-hatching each side, 8 to 10 minutes per side (see Cook's Note). The wedges should have a cooked exterior and a crunchy interior, with dark grill marks and charred edges.
Meanwhile, mix together the buttermilk, mayonnaise, horseradish, mustard, 1/2 teaspoon salt and 1/4 teaspoon pepper in a medium bowl. Reserve.
Gently toss together the parsley, cider vinegar, carrot, scallion, a drizzle of olive oil and salt and pepper to taste in a medium bowl. Adjust with more vinegar, oil, salt or pepper as needed.
Cut off the core from the cabbages and place the wedges on a platter. Drizzle with about half the buttermilk dressing; serve the rest on the side. Scatter the corned beef cracklings and carrot-parsley salad on top.
To make cross-hatch grill marks, arrange items on a grill at a 45-degree angle, then a quarter of the way through the cooking time, give the items a 90-degree turn. Flip halfway through cooking and repeat on the other side.
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