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Heat oven to 425°F. Poke potatoes a few times with a fork, place them on a microwave-safe plate, and microwave on high until very hot to the touch and starting to soften at the surface,
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2 to 3 minutes. Halve the potatoes and set aside.
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In an ovenproof skillet over medium-high heat, heat the oil until hot but not smoking, about 2 minutes. Generously season chicken with salt and pepper. Add the chicken to the pan, skin side down; cook for 5 minutes or until the skin is well browned and crispy. Turn over the chicken. Add the potatoes, cut side down, and roast in the oven for 8 minutes. Add the garlic, ensuring the cloves are immersed in the pan juices, and roast until the chicken is cooked through, 3 to 4 minutes more. Remove the skillet from the oven. Remove and reserve the potatoes and chicken, keeping the chicken skin side up.
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Return the pan to the stovetop and heat over medium-high heat. Add the broth, lemon slices, and oregano and bring to a boil. Boil for 4 minutes to reduce and thicken the broth; season with ¾ teaspoon salt and pepper to taste, then stir in the spinach until wilted. Swirl in the butter and return the potatoes and skin-side-up chicken to the pan along with any collected juices. Simmer for 4 minutes more. Serve.