To a large pot, heat two tablespoons of olive oil over medium heat. When the oil is hot, add the diced carrots and shallot. Cook until the carrots have softened and the shallots turn translucent, about 5 minutes. Add the garlic, spices, red pepper flakes, tomato paste and ground lamb, using a wooden spoon to break up the lamb in to crumbles. Sprinkle the lamb with a few pinches of salt and cook, stirring frequently, for about 5-7 minutes, until the meat has browned.
Next, pour in the can of tomatoes. (If you're using a can of cherry tomatoes, take the back of your spoon and crush them.) Add 1/2 cup (you can eyeball this measurement) of water to the tomato can, swishing it around so you get all of the remnants of tomato juice; pour the tomato water into the pot and stir. Bring the tomato sauce to a slight simmer and then turn the heat down to low. Cover the pot and cook for 20 minutes, stirring frequently. At the 20 minute mark, stir in the kalamata olives, balsamic vinegar, Parmesan-Reggiano and a splash or two of water, if it's cooked down a bit too much. Cover the pot once more and cook for an additional 20 minutes, until the sauce is fragrant and has thickened slightly. (If at any time the sauce becomes too thick, you can simply add a bit more tomato paste and a 1/4 cup of water or veggie stock.) Right before tossing with the pasta, add the butter and mix until melted.
Meanwhile, boil some salted water and cook the paparadelle per the box's instructions. (Mine told me 10 minutes exactly.) Toss the pasta with a few tablespoons of sauce and serve with a few spoonfuls of bolognese on top. Garnish with more Parmesan-Reggiano and a sprinkle of Italian parsley