Season the birds inside and out with salt and pepper.
Heat the olive oil in a big saucepan for which you have a cover, over medium-high heat. Saute the crushed garlic cloves until they brown around the edges, then remove and reserve them.
Add the birds to the pan and brown them all over. Remove the birds. Pour out the oil and wipe the pan clean with a wad of paper towels. Replace the birds in the pan.
In a second saucepan, add the steak sauce to the chicken stock and bring it to a light boil. Pour it over the birds in the pan, and return to medium heat. Add all but two of the garlic cloves, plus the honey, thyme, and a little salt and pepper. Bring it up to a boil, then lower to a simmer. Cover the pan and cook for about 40 minutes for the quail, about 1 hour and 10 minutes for the Cornish hens.
Place the birds in a serving dish deep enough to contain all the sauce. Remove all the other ingredients (except the thyme, which you discard) to a blender or food processor and puree. Add 1/4 cup of water if needed to help things along.
Strain the sauce into a clean pan and bring to a boil. Add the parsley, the zest, and the fish sauce. Chop the two cloves of garlic left over from the first step and add them. Cook for a minute, then add the tomatoes. When they soften, serve the sauce over the birds.