In a saucepan, bring 1 cup water to a boil. Add the salt and stir in the rice. Cover, reduce the heat, and simmer over low heat until the water has been absorbed and the rice is fully cooked, 15 to 20 minutes. Transfer the rice to a bowl and set aside. Melt the butter in a medium saute pan over medium heat. Add the onion and cook until tender and translucent, 8 to 10 minutes. Stir in the currants (drained) and allspice and cook a few minutes more. Stir the onion mixture, pine nuts, and lemon zest into the cooked rice. Season with salt and pepper. Let cool.

Heat the oven to 400 degrees F. Make a basting sauce by melting the butter and stirring in the allspice, garlic, and lemon juice. Season with salt and pepper and set aside.

Spoon the cooled stuffing into the cavity of the hens. Set the hens on one large or two small racks in a large shallow roasting pan or large rimmed baking sheet. Spoon or brush half of the lemon butter over the birds. Sprinkle with salt and pepper and roast, basting occasionally with the butter, until the juices run clear when the leg is pierced with a skewer, 60 to 70 minutes. The internal temperature of the stuffing should be 160 degrees F. Remove the birds and racks from the pan. Spoon off and discard the fat; pour any juices into a separate container. If there are nicely browned juices in the bottom of the pan, deglaze with a little broth or water and add to the other juices. Serve one bird per person with some of the pan juices, if you like.