In a medium saucepan over medium heat, melt 2 tablespoons of the butter. Add the leeks and cook until soft, 10 to 12 minutes. Add the wild rice, chicken broth, water, salt, and pepper. Bring to a boil, reduce the heat to low, cover, and simmer gently without stirring until the rice is just tender, 40 to 45 minutes. If the rice is done and liquid remains, pour off the extra. Set aside. Meanwhile, melt 1 tablespoon of the butter in a large frying pan over medium heat. Add the mushrooms, parsley, thyme, and sage to the pan. Season with salt and pepper and cook, stirring occasionally, until the mushrooms are soft and have released their liquid, 3 to 5 minutes. Continue to cook, stirring occasionally, until the liquid has evaporated and the mushrooms are dry, 3 to 5 minutes. Stir the mushrooms into the rice and set aside to cool; you should have about 2 cups of stuffing.

Heat the oven to 425 degrees F. Season the cavity of each hen with salt and pepper. Make sure the stuffing is thoroughly cool and pack the cavity of each hen loosely with about 2/3 cup stuffing. With kitchen twine, tie the legs together. Tuck the wings underneath. Arrange the birds breast side up on a wire rack set in a shallow roasting pan (or two). Melt the remaining 2 tablespoons butter and use half to brush over the hens. Sprinkle them with salt and pepper. Roast for 20 minutes. Brush with the remaining melted butter. Continue to roast until the juices run clear when you prick the thickest part of the thigh and an instant-read thermometer inserted in the thigh registers 170 degrees F, another 25 to 30 minutes. Transfer the hens to a platter, tent with foil, and let stand for 10 min. before serving.