Heat the olive oil in a large pot over medium-high heat. Add half of the lamb and cook, turning, until browned, 8 to 10 minutes; remove to a plate. Repeat with the remaining lamb; remove to the plate.
Add the tomato paste to the pot and cook, stirring, until well combined, about 1 minute. Add 6 cups water and 1 teaspoon salt, then return the lamb to the pot. Cover and simmer, stirring occasionally, until the meat is very tender and falling off the bones, 2 to 2 1/2 hours.
Remove the lamb using tongs and transfer to a large bowl. Shred the meat, discarding the bones and excess fat, then return the meat to the pot. Simmer, uncovered and stirring occasionally, until the sauce thickens, 45 to 60 minutes.
Meanwhile, bring a separate large pot of salted water to a boil. Add the pasta and cook as the label directs. Reserve 1 cup cooking water, then drain.
Add the pasta and 1/2 cup reserved cooking water to the sauce. Cook, stirring, until coated, about 1 minute, adding more pasta water as needed to loosen. Remove from the heat and stir in the mint and lemon zest; season with salt and pepper. Divide among bowls and top with more mint and lemon zest.