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Heat olive oil in a large skillet over medium heat. Add chicken, and season with salt, pepper, and oregano. Cook until browned, and no longer pink in the middle, about 8 minutes per side. Remove from skillet and set aside.
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Melt the butter in the same skillet over medium heat. Add minced garlic and cook for about 1 minute. Stir in cherry tomatoes and season with salt and pepper. Cook until tomatoes begin to burst, then add spinach and cook until spinach wilts.
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Stir in heavy cream and Parmesan and bring to a simmer. Reduce heat to low, simmering until sauce is slightly reduced and thickened a bit, about 3 minutes. Add chicken back to the pan, and cook until heated through, 6 to 7 minutes.
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Serve with an additional sprinkle of Parmesan on top, and crusty bread to dip in the sauce.