Bring a pot of water of lightly salted water to a boil. Add the green beans and cook until tender, about 6 minutes. Drain and transfer to an ice water bath to stop the cooking. Drain and set aside.

In another pot, place the potatoes and 1 teaspoon of the salt in a large saucepan, cover with water by 1-inch, and bring to a boil. Reduce the heat to a simmer and cook until tender, 20 to 25 minutes. Drain the potatoes and return the potatoes to the pot. Set over low heat and add the butter, 1 teaspoon of the salt, 1 teaspoon of the black pepper, and 1/4 cup of the milk. Mash with a potato masher over medium-low heat to incorporate the ingredients and until the mixture smooth but slightly lumpy, 3 to 4 minutes. Set aside.

In a large, heavy saute pan or skillet, heat the oil over medium-high heat. Add the lamb or beef, the remaining 2 teaspoons salt and 2 teaspoons freshly ground black pepper, and cook, stirring, until well browned, about 6 minutes. Add the onions and the celery, and cook, stirring, until the vegetables begin to soften, about 2 minutes. Add garlic and cook until fragrant, about 30 seconds. Add the green beans, and cook for 5 minutes, stirring occasionally. Add the tomatoes and cook, stirring for 5 minutes. Add the remaining 1/2 cup milk and cook, stirring, for 4 minutes. Remove from the heat and reserve for later use.

Spread half of the potatoes in the bottom of a 9 by 13-inch baking dish. Pour in the meat mixture, and top with the remaining mashed potatoes. Sprinkle with the cheese and bake until bubbly and golden brown, about 1 hour.

Remove from the oven and serve hot.