Season the duck legs with salt and pepper. In a Dutch oven, heat the olive oil over medium-high heat. Add the duck legs, skin-side down, and sear until the skin is golden brown, about 5 minutes per side. Remove the duck and set aside.
In the same pot, add the onions, carrots, and celery. Cook over medium heat until the vegetables are softened, about 5 minutes. Stir in the garlic and cook for another minute.
Pour in the red wine, scraping the bottom of the pot to loosen any browned bits. Allow to reduce by half, then add the crushed tomatoes and chicken stock. Return the duck legs to the pot, and add the rosemary, thyme, and bay leaf. Bring to a simmer, then reduce the heat to low, cover, and cook for 1.5 to 2 hours, or until the duck meat is tender and easily pulls away from the bone.
Remove the duck from the pot and let cool slightly. Discard the skin and bones, and shred the meat using two forks. Return the shredded duck to the pot and stir into the sauce. Season with additional salt and pepper to taste.
Bring a large pot of salted water to a boil. Cook the Bigoli pasta according to package instructions until al dente. Drain, reserving a cup of pasta water.
Add the cooked Bigoli to the ragu, tossing to coat the pasta evenly. If the sauce is too thick, add a bit of the reserved pasta water to reach the desired consistency.
Plate the pasta and ladle generous amounts of duck ragu over the top. Finish with a sprinkle of grated Parmesan cheese and serve immediately.