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Slice off the tip of the garlic head to expose the cloves, leaving the root end intact. Set aside.
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In a Dutch oven or a large pot over medium-high heat, add the oil and 2 tbsp butter. Add onion, carrots and 1 tsp salt, then cook for 8 to 10 minutes, until vegetables are soft and golden. Add the chicken, bay leaves, pepper and 1 tsp salt. Cook, stirring occasionally, for 5 minutes, until the chicken is golden on all sides.
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Increase the heat to high and add the wine. Cook, scraping up the browned bits from the bottom of the pot, for 3 to 5 minutes, until most of the wine is evaporated. Add the broth and rice and place the garlic head, sliced side down in the center of the rice. Bring to a boil, then reduce heat to LOW and cover.
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Simmer for 15 minutes, until the rice is fully cooked.
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Remove pot from the heat and remove and discard the bay leaves. Take out the garlic head. Discard or squeeze out cloves and stir into the rice. Add the remaining 2 tbsp butter, Parmesan cheese and parsley. Stir until the butter is melted. Season with salt and pepper if needed.
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Serve with more Parmesan and parsley.