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Preheat the oven to 350 degrees.
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Melt the butter in a large skillet over medium heat.
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Add the onions, celery, and carrots. Cook uncovered, until the onions are soft and translucent. About 7 minutes.
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Add the minced garlic and cook for an additional minute.
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Add the turkey, corn, and green beans (or peas) in addition to the salt, pepper, and thyme.
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Gently stir in the gravy: If preparing a packet of gravy, cook that separately according to package instructions. Once thickened, add it to the skillet.
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Once the mixture is heated through, pour it into a one-quart casserole dish, spreading it into one even layer.
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If using leftover mashed potatoes, heat them a bit to make it easy to stir and spread. Add half of the cheese to the potatoes and stir to combine.
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Spread the potatoes in an even layer over turkey mixture and top with the remaining cheese.
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Heat in the oven for 30 minutes, uncovered. Serve warm, and enjoy!