I got my inspiration for this recipe from a similar one in Paula Wolfert's great book, Mediterranean Grains and Greens. Paula bakes her pigeons, however, and I prefer them grilled. My friend Soraya says grilled pigeons are all over the place in Cairo, where she used to live, so there you have it. What the hell is green wheat? The Egyptians call is freekeh, and it's slightly unripe wheat that's been parched (slightly burned) and dried. It cooks really fast and has a great smoky flavor. But... you can only get it in specialty shops, or online. If you want to go an easier route, use bulgur wheat, which is readily available in most supermarkets. Still too hard? Use rice.
Mix all the spices for the rub together. Coat the pigeons or doves with oil, then dust with the spice rub. Put the birds in a covered container in the fridge for 3 hours, or up to overnight. An hour before you plan to cook them, take the birds out of the fridge and let them rest on the counter to come to room temperature.
Meanwhile, make the green wheat salad. Bring the broth to a boil and add the freekeh or bulgur. Stir, let it come back to a boil, then cover and turn off the heat. It should be ready in 30 minutes or so.
Put the olive oil in a frying pan and turn the heat to medium-high. When the oil is hot, saute the shallots for a minute or two, or until they just start to brown. Add the garlic and bell pepper, saute for another minute, then turn off the heat.
Get your grill hot. Charcoal is your best bet here, but any sort of open flame will work. When the grill is hot, drizzle a little more olive oil on the pigeons and set them on the grill, breast side up. Let them cook like this for 10 minutes, making sure they don't get burned to a crisp. You want to do most of the cooking with the breast side up. Flip the birds and set them breast side down on the grill, making sure the middle of the breast slips between the grill grates -- this helps the birds lie flat. If you want to get better contact with the grill, place a brick on the pigeons while they cook on this side. Grill for 3 to 5 minutes, depending on how hot your fire is and how well done you like your birds; I prefer my doves and pigeons to be pink inside. Take the birds off the grill and let them rest, breast side up.
To finish the salad, mix the green wheat or bulgur into the frying pan and add the olive oil, parsley, salt and pepper, plus lemon juice to taste. Serve alongside the pigeons.