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.
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.
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.
Serve with an additional sprinkle of Parmesan on top, and crusty bread to dip in the sauce.
Nutritional information is provided as a courtesy only and should be viewed as an estimate. Ingredients can vary and The House on Silverado makes no guarantees to the accuracy of this information.