Pulse the garlic, ginger, 1 tablespoon olive oil and the paprika in a food processor to make a paste. Heat the remaining 2 tablespoons olive oil in a medium skillet over medium-high heat. Add the garlic-ginger paste, 1 teaspoon salt and a few grinds of pepper; cook, stirring occasionally, until the paste begins to fry slightly, about 5 minutes. Transfer to a bowl; let cool.
Make about sixteen 1-inch-deep slits all over the leg of lamb with a paring knife. Stuff the chopped mint and some of the garlic-ginger paste into the slits. Rub the remaining paste all over the outside of the lamb. Wrap the lamb in plastic wrap and refrigerate at least 4 hours or overnight.
Bring the lamb to room temperature 30 minutes before roasting. Preheat the oven to 375 degrees F. Unwrap the lamb, season with salt and pepper and place on a rack in a roasting pan. Pour 1 cup chicken broth into the pan. Transfer to the oven and roast until the lamb starts browning, about 30 minutes. Meanwhile, combine the remaining 2 cups chicken broth, the vinegar and sugar in a medium saucepan and bring to a boil; remove from the heat and set aside.
Remove the lamb from the oven; baste with the broth mixture. Continue roasting, basting every 30 minutes, until the lamb is golden brown and a thermometer inserted into the thickest part registers 130 degrees F, about 1 1/2 more hours. Transfer to a cutting board and let rest 20 minutes before slicing.
Meanwhile, spoon off any excess fat from the pan juices; add the pan juices to the remaining broth. Heat the broth mixture over medium heat, stirring occasionally, until reduced by about one-third, about 5 to 7 minutes. Top the lamb with mint and serve with the broth mixture.