From my Greek mom, who makes an avgolemono pretty close to what I've had in restaurants though it's not yellow. I just got off the phone with her about this and I'm translating from Greek so it's far from exact but here is what she told me:
Cook chicken pieces in water to cover until it's done (she didn't really specify how much water)
Remove chicken pieces, remove meat and discard skin and bones
Strain broth
Add rice - about a cup maybe
Cook until rice is done - about 30 minutes and she said something about how the broth/rice should look "cloudy" (I think that's how it translated from Greek)
Remove from heat and let it cool 5-10 minutes
Whisk 4 eggs well with a fork
Whisk juice from 1 1/2 - 2 lemons into eggs
Temper egg-lemon mix by whisking in a bit of the hot rice/broth (this may prevent the egg-drop soup effect)
Slowly whisk the eggs and lemons back into the soup
Bring soup up to barely a simmer
Add in the meat from the chicken - as much or as little as you like
Season - add more lemon juice if it needs it.
Done
I've not made this myself yet, so let me know how it turns out.