Heat the oven to 375 degrees and use 1 tablespoon of the butter to grease a 1 1/2-quart baking dish. Rub the sweet potatoes with 1 tablespoon of the olive oil, and pierce each one a few times with a sharp fork. Wrap them in aluminum foil and bake until tender, about an hour.
Meanwhile, heat the vegetable oil in a heavy saucepan over medium-high heat. Add the ground lamb and a couple pinches of salt and brown, stirring frequently, 3 to 5 minutes. Remove the meat to a plate with a slotted spoon and turn the heat down to medium-low.
Add the onion and carrot to the pan and cook, stirring occasionally, until soft, about 5 minutes.
Stir in the tomato paste and cook for 2 minutes. Stir in the flour and cook for another 2 minutes. Stir the meat back into the pan.
Add the stock or water and a couple more pinches of salt and raise the heat under the pan to bring the liquid to a boil. Cover the pan, lower the heat and simmer very gently, stirring every once in a while, until the mixture is thick but still a bit saucy, 10 to 15 minutes. Add more salt if necessary, and set this aside to cool a little while you finish the potatoes.
When the potatoes are tender, remove them from the oven and turn the oven temperature up to 425 degrees. Carefully unwrap the foil and slit the potatoes down the middle, Scoop out the flesh with a spoon, putting it into a heatproof bowl. Using a wooden spoon, beat in the remaining 3 tablespoons olive oil until you have a smooth, silky mash. Add salt to taste.
Transfer the meat to the buttered baking dish, smoothing the top. Spread the mashed sweet potatoes over the top, using a spatula to gently work it out to the edges of the dish and cover all of the meat.
Dot the top of the potatoes with the remaining tablespoon of butter and set the dish on a baking sheet with high sides to catch any spillover. Bake the cottage pie until it is heated through, and the top begins to brown, 10 to 15 minutes. (You can use the broiler at the end to get a little more color on the potatoes if you'd like.) Let cool for a few minutes before serving.