In a large bowl, mix the yogurt, ginger, garlic, tandoori masala, and 1/2 teaspoon salt.
Rinse quail and pat dry. Using poultry shears or a sharp knife, cut out the backbone of each quail. Turn birds skin side up and press with the palm of your hand to flatten; a few bones will crack.
Turn birds in yogurt mixture to coat. Cover and chill at least 30 minutes or up to 1 day.
To barbecue, place quail on an oiled grill over medium-hot coals or over medium-high heat on a gas grill (you can hold your hand at grill level only 3 to 4 seconds). Close lid of gas grill. Cook quail, turning as needed, until skin is well browned and meat at breastbone is still pink (cut to test), 12 to 14 minutes total. Season to taste with salt.