Make the peppers: Combine 2 1/2 cups water, the rice, currants and 1/4 teaspoon salt in a medium saucepan. Bring to a boil, then reduce the heat to low; cover and simmer until the water is absorbed, about 45 minutes. Let stand, covered, 10 minutes. Fluff with a fork, then transfer to a large bowl to cool, about 20 minutes.
Meanwhile, preheat the oven to 350 degrees F. Spread the walnuts on a baking sheet and roast until lightly golden, 6 to 8 minutes; let cool, then finely chop. Pile the dill, parsley and scallions on a cutting board and finely chop.
Cut off the top of each bell pepper and reserve the tops; discard the seeds and white membranes. Put the turkey in a large bowl; add 1 cup mozzarella, 1/4 cup feta, the rice mixture, walnuts, all but 2 tablespoons of the chopped herbs, the garlic, lemon juice, lemon zest, 1 1/2 teaspoons salt and 1/2 teaspoon pepper. Gently mix with your hands. Stuff each bell pepper evenly with the turkey-rice mixture; cover with the pepper tops.
Make the sauce: Whisk 3 cups water, the tomato puree, brown sugar, olive oil and cinnamon stick in a wide, deep saucepan. Arrange the stuffed peppers upright in the sauce. Bring the sauce to a simmer over medium-high heat. Cover and reduce the heat to medium low; simmer until the peppers are soft and the filling is cooked through, 50 to 60 minutes.
Preheat the broiler. Carefully remove the peppers from the sauce and transfer to a flameproof baking dish; remove the tops and set them aside in the baking dish. Toss the reserved 2 tablespoons chopped herbs with the remaining 1 cup mozzarella and 1/4 cup feta in a small bowl and sprinkle over the filling. Broil until melted, about 2 minutes. Replace the pepper tops. Season the sauce with salt and pepper. Serve the peppers in shallow bowls with the sauce.