Since this is a very popular question, asked almost as many times as "how do I breed my own Cornish X chickens at home", I thought I would give my opinion on how to do it.
This would not be a project for anyone in the city because it would take probably hundreds of generations and thousands of birds.
The reason it would take so many birds and generations is because there is no existing "silent" gene for chickens. It's not like starting a new variety that lays blue eggs. There is already an existing gene for blue eggs that could be worked with, so it is not difficult to develop a new blue egg chicken (as long as blue eggs are the only criteria).
Add in some intensive record keeping. Which birds are the most quiet? Who are the parents? Does one rooster or one hen produce offspring with softer voices than the others? Each generation, for generation after generation, select the birds that have the least loud voices. Also you want less frequent noise, but the quieter voice is the key.
Be aware that if you are selecting for quiet voices, you will probably lose egg laying and meat qualities unless you also breed for those at the same time.
I have a difficult time imagining this project ever getting done, because the people with the time, space, and money to do it don't need a quiet chicken. If they have room for hundreds of chickens, then noise is not an issue for them.
For people inside city limits who want quiet chickens, it would probably be cheaper and easier to just buy a place in the country than it would be to breed a quiet chicken.
Exhibition quality Blue Swedish Ducks and Gray Saddleback Pomeranian Geese, Hatching eggs available in late winter and spring. NPIP