It might sound hard to make this sauce if you are not in the desert Southwest, but a lot of the esoteric ingredients are available online, such as the chiltepin chiles and prickly pear syrup, or you can easily substitute for them by using a habanero or other really hot chile and some good honey. It will be a little different, but still good.
1/2cuphoney, mesquite bean syrup, or prickly pear syrup
1to 2 teaspoons ground dried hot chiles,or hot sauce to taste
Salt to taste
Make the sauce first. Heat the butter or lard over medium heat, then add the grated onion and sauté until it turns translucent, about 3 to 4 minutes. Add the remaining sauce ingredients, stir well and simmer slowly for 30 minutes or more. I always pour the sauce into a blender and puree it, but you can serve it chunky if you want.
Flatten the quail. Remove the backbones with kitchen shears by cutting along either side of the spine. Put the quail breast side up on a cutting board and press to flatten. If you want to be fancy, carefully snip out the ribs and the curved saber bones near the wishbone. Salt the quail and toss them in the vegetable oil.
Grilling. Get your grill hot, but leave an open space with no coals, or with one gas burner left off. Lay your quails, breast side up, on the cool part of the grill and cook with the grill lid down for 10 minutes. Paint the breast side with the sauce, and repeat this two more times until you've cooked the quail for 30 minutes. Check the doneness, either by inserting a thermometer into the thickest part of the breast (it should read 150°F to 155°F), or by testing where the legs meet the thighs: They should want to come apart when wiggled, but not fall apart. You might need another 10 or even 20 more minutes to get to this point, depending on how hot your barbecue is.
To finish. Turn the quail over and paint the cooked side with sauce. Grill the breast side for 2 minutes with the lid up, then turn over again and paint with the sauce one more time. Cover the grill and cook for a final 2 minutes. Take off the grill and paint with sauce one more time.