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Preheat oven to 375F. Pay the chicken dry and season liberally with salt and pepper.
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In a large skillet over medium heat, heat the oil until shimmering. Add the chicken, skin side down. Cook undisturbed for 5 to 7 minutes, until the skin is golden and crisp. Flip over and cook for anohter 4 minutes.
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Transfer the chicken to a baking dish and place in the oven. Bake for 15 minutes or until the internal temperature reaches 165F.
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While the chicken is baking, prepare the mushrooms. Melt the butter in the hot skillet. Add the mushrooms and saute until golden brown, about 5 minutes. Add the rosemary, garlic, salt, and pepper and cook for 1 more minute. Stir in the cream and simmer until thickened.
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Add the chicken and any accumulated juices back to the pan. Spoon the mushrooms and sauce over the thighs as you serve them.