To prepare the hens, sprinkle with 1/2 teaspoon salt and 1/4 teaspoon pepper. Place the onion and next 5 ingredients (onion through allspice) in a food processor; process until smooth. Combine hens and onion mixture in a large zip-top plastic bag. Seal bag; marinate in refrigerator 1 hour, turning bag occasionally. Remove the hens from the bag; discard marinade.
Preheat oven to 400°.
Place hens on a broiler pan coated with cooking spray; bake at 400° for 35 minutes or until thermometer registers 180°.
To prepare the bulgur, soak currants in a bowl of water 15 minutes. Drain; set aside. Bring broth to a boil in a medium saucepan. Add the bulgur, 1 1/2 teaspoons cinnamon,1/2 teaspoon salt, and 1/4 teaspoon pepper; cover, reduce heat, and simmer 15 minutes or until liquid is absorbed. Stir in currants, pine nuts, and butter.