Heat the olive oil in a large Dutch oven over medium high heat. Saute the carrots, onion, and celery for 5 minutes or until softened, stirring often. Add the garlic and saute, stirring often, for 1 to 2 minutes or until fragrant. Add the lamb, coriander, fennel, cumin, rosemary, and thyme; season with salt and pepper. Cook for 5 to 7 minutes until the lamb browns. Spoon off any extra grease (I removed a good 1/4 cup or so!).
Add the red wine and let cook until evaporated, about 5 minutes. Add the tomatoes and their juices, along with the stock and bring to a boil. Reduce heat and cook over moderately low heat, partially covered, for about 35 minutes or until the mixture begins to thicken. Stir in the Balsamic vinegar and season with salt and pepper. Cover and remove from the heat.
In a large pot of boiling salted water, cook the pasta according to package directions, until al dente. Drain, shaking well. Add the pasta to the sauce, tossing to combine. Serve the pasta in bowls, topped with grated Pecorino (or a dollop of ricotta) and chopped fresh mint.
NOTE: This ragu is especially delicious if made the day before. Store in an airtight container in the refrigerator and reheat gently on the stovetop, stirring in an additional 1/2 cup chicken broth to loosen it up a bit. Boil the pasta and serve as instructed above.