It just takes some record keeping.
You'll need to keep track of the parentage. When you get an exceptional tasting bird, you add full siblings of that bird into your breeding flock.
Livestock breeders keep track of all sorts of things: depth of back fat, size of rib eye, number of eggs laid every year, pounds of growth per pound of feed.
Just be aware that what the chicken is eating affects the taste, texture, and tenderness of the meat, so you will want to have your test chickens raised in exactly the same way.
Another thing you will want to keep improving is size of the breast meat. That can be easily determined without killing your bird, so you want to keep the birds with plump breasts for your breeding flock.
If you really want to do this project, you could come up with a chicken with high demand once it becomes known that your birds have exceptional flavor.
(I am assuming you are talking about chickens, but any poultry can be bred that way, with an emphasis on taste)