Season the lamb with salt and pepper. In a 5-to 6-quart Dutch oven, heat the oil over medium- high. Working in batches, brown the lamb on all sides, adding more oil as needed and reducing the heat if the lamb begins to burn, about 5 minutes per batch. Transfer the lamb to a plate. Pour all but 1 tbsp. of fat from the pot.
Heat the remaining drippings over medium. Add the onions and carrots; season. Cook, stirring often, until the vegetables just begin to brown, about 5 minutes. Add the garlic and rosemary; cook, stirring often, until fragrant, about 1 minute. Add the tomatoes. Swish 1 cup of water in the tomato can; add the water to the pot. Stir in the cream and return the lamb with any juices to the pot. Cover and reduce the heat to low. Simmer, stirring occasionally and adding water by 1/4 cupfuls if too dry, until the lamb is tender, 1 1/2 to 2 hours; season.
Meanwhile, in a medium saucepan over medium, bring 6 cups of water and 1 tbsp. salt to a boil. Gradually add the polenta, whisking to prevent lumps. Reduce the heat to low and simmer, stirring often until smooth and thick, 20 to 30 minutes for regular polenta and about 5 minutes for instant polenta. Stir in the butter.
Spoon the polenta into shallow bowls and ladle the stew on top. Garnish with the chopped basil.