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Prepare artichokes.
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In a large deep pan, put quartered artichoke hearts in a steamer basket over boiling water. Steam until the leaves of the artichoke pull away from the heart and the heart is easily pierced with a knife. This will take 15-20 minutes. Remove from pan and drain the artichokes.
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Whisk lemon zest, lemon juice, balsamic vinegar, olive oil, garlic, sea salt, pepper and three-quarters of the chopped parsley in a large bowl. Add the steamed artichoke quarters to the bowl of marinade. Let the artichokes sit in the marinade for at least 30 minutes and up to 3 hours to absorb the flavors in the marinade.
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Remove the artichoke quarters from the marinade and, over a hot bbq grill, cook the artichoke quarters until grill marks appear. (Watch the artichokes carefully while grilling. You want them to be charred but not burned to a crisp.) Flip the quarters and grill the other side. This will take 3-5 minutes. You want the artichokes to be completely tender. Transfer the grilled artichokes to a platter. Drizzle with some of the remaining marinade.
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Grill the lemon quarters flesh side down on your bbq grill until you see grill marks and the sugars from the lemons begin to caramelize.
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Arrange the grilled lemon quarters on top of the artichokes. Arrange some Kalamata olives artfully on top of the dish. Garnish with remaining parsley and sprinkle salt and pepper on the dish. Serve immediately.