In a food processor, add the carrot, leek, onion, and garlic and pulse until finely chopped. (Alternatively, you chop the vegetables very finely by hand). Set aside.
In a large dutch oven or stockpot, heat 3 tablespoons of the olive oil over medium-high heat. When the pot is very hot, add the duck legs in one layer and sear on all sides until golden brown, about 8 minutes total. Transfer to a paper towel-lined plate and set aside. If there are any burned black bits in the bottom of the pot, carefully wipe them away with a paper towel (otherwise, any browned bits should be left as is).
In the same pot, heat the remaining ½ cup of the olive oil over moderate heat and add the finely chopped vegetables. Cook, stirring frequently, until softened, approximately 5-8 minutes. Add the salt, pepper, tomato paste, rosemary sprig and bay leaves. Stir to combine and cook for another 2-3 minutes. Add the red wine and simmer until most of the liquid has evaporated. Stir in the canned tomatoes, and gently break them up using a wooden spoon. Add the chicken stock and stir until combined. Return the duck legs to the pot, adjusting them so they are all submerged in the sauce, and reduce the heat to low. Cook, partially covered, for 1 hour, stirring occasionally to ensure even cooking.
After an hour, remove the duck legs from the pot and transfer to a chopping board. Leave the sauce simmering on low heat, uncovered. Using two forks, remove the duck meat from the bones and roughly shred. Discard the duck skin and bones and return the shredded meat to the sauce. Continue to simmer the sauce over low heat, uncovered, for about 10 minutes, or until it reaches your desired consistency. Remove and discard the sprig of rosemary and bay leaves. Taste the sauce and adjust the seasoning as desired.
This ragu is best served with a pasta such as tagliatelle or pappardelle and some good-quality grated parmesan cheese on the side.
Not into duck? You can substitute it with a pound of ground beef.