Process the garlic and ginger to a paste in a food processor. Add the anise, saffron, paprika, cayenne, and lemon juice and pulse to combine. Transfer the mixture to a bowl, add the chopped cilantro and green onions, and whisk in the olive oil. Season to taste with cayenne and black pepper; the marinade should be spicy!
To butterfly the poussins, insert a heavy, sharp knife at the neck cavity and carefully cut down one side of the backbone. Cut along the other side of the backbone to separate it from the body. Pull open the sides of the poussin, turn it skin side up, and press firmly on the breast with the heel of your hand to flatten it. Fold the wings back against themselves so they lie flat. Trim the excess skin at the neck end.
Place the butterflied poussins in a shallow nonaluminum container; cover with the marinade and turn the poussins to coat. Cover and refrigerate overnight.
Let the poussins warm to room temperature. Heat the grill or broiler. Remove the poussins from the marinade and reserve the marinade. Sprinkle the birds lightly with salt. Grill or broil, skin side away from the flame, about 5 minutes, basting occasionally with the marinade. Turn and continue cooking and basting until the juices run clear when the leg is pierced with a skewer and the skin is golden and slightly crisp, about 5 more minutes. Serve hot with lemon wedges.