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Leaving the core intact, cut each radicchio in halves or quarters if large heads.
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Brush with half of the oil.
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Sprinkle with half of the sea salt and half of the freshly ground pepper.
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Grill the radicchio on high heat, uncovered, turning often.
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The colour will change and ideally a nice 'char' will form around the edges.
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Remove and set aside.
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Whisk all but the cheese with the remaining olive oil.
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Toss radicchio pieces in a bowl.
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Marinate for 10-15 minutes. Drain and reserve marinade
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Top with shavings of Pecorino Romano or Parmesan.