The dual purpose, larger bodied breeds, like the White Rock, Delaware, etc, were used for this purpose on the homestead. However, the females won't lay until they are 22, sometimes 24 weeks old. The cockerels won't becomes fertile roosters and master their breeding techniques until they are about the same age. Most people, who want tender meat, don't allow the birds to reach that age. Most dual purpose birds are processed between 15-18 weeks of age for maximum feed to weight ratio and cost effectiveness, as well as meat that is reasonably tender.
When the mature birds have sexual hormones flowing through them, the texture of the meat changes.
The typical CX meat bird is somewhat difficult to bring to sexual maturity. They are typically processed at 7-8 weeks of age. Even if you can/could limit feed enough to "breed" the CX, the result would not be as satisfactory as you might suppose. The CX is a complicated 4 way cross of grandparent stock, selectively bred for 50 years.
I hope you can see the challenge of doing what you are asking. Not impossible, just challenging.