Heat the olive oil and 2 tablespoons of the butter in a large skillet over medium heat. Add the onion, carrots, celery, mushrooms, garlic, half the salt, and oregano. Cook until vegetables are tender, about 10 minutes. Stir in the tomato paste and cook until mushrooms are soft and tomato paste has turned brick red, about 8 minutes more. Stir in the beef, the broth, the remaining salt, the Worcestershire, and some pepper, breaking up any large clumps of meat, cook until the meat is no longer pink, about 3 minutes.
Transfer the meat and vegetables to a 2-quart oval casserole dish and spread the mashed potatoes over the top, leaving a 1/4-inch boarder around the edge. Make a decorative pattern on the top of the potatoes, if desired. Sprinkle with cheese, if using, and dot with the remaining tablespoon of butter. Bake until potatoes brown and the juices bubble around the edge, about 40 minutes. Let cool for 10 minutes before serving.
Put the potatoes in a saucepan with cold water to cover by about 1-inch and add the salt. Bring to a simmer over medium-high heat, uncovered, until the potatoes are tender but not mushy, about 15 minutes. At the same time, combine the milk and butter in a small saucepan and warm over low heat until the butter is melted. Remove from the heat and cover to keep warm.
Drain the potatoes and return to the saucepan. Toss the potatoes over medium heat until dry. Add the heated milk mixture and mash the potatoes until just slightly chunky, then stir in the parsley and season with salt and pepper, to taste.
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