Place the chicken in a large stockpot. Cover generously with water, and add the whole onion, whole clove of garlic, and a teaspoon of salt.
Bring to a simmer over medium heat, and then turn down to low and cook gently, lid ajar, for 60 - 90 minutes. When the chicken is cooked through and tender, but not yet falling apart, use a slotted spoon to remove it and the onion and garlic to a colander over a large bowl. Taste the broth for salt and adjust as needed.
Rinse the rice and add it to the broth. Continue to simmer, stirring it periodically.
As soon as the chicken is cool enough to handle, chop or shred the meat into small pieces, and stir it into the simmering broth. Discard the onion and garlic.
When the rice is fully tender and you are ready to serve, finish the soup. Squeeze the juice from half the lemon into a medium bowl. Beat in the egg yolks, then beat in about a cup of the hot broth. Stir the tempered mixture into the hot soup; keep stirring for a minute or two, or until heated through and slightly thickened, but do not boil.
Taste again for salt and serve immediately, along with the remaining lemon half, in wedges, on the side.