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Mutts do have a purpose for heterosis, or hybrid vigor, and this can get you somewhere for meat concerns, if for no other reason that if you can kill it you can eat it. However, laying, like dairy, is distinctly linked to the genetics of the breed and is maintained carefully through controlled in-breeding within the breed. Certain crosses are productive in the first generation, especially, and specifically, if you use a cock from a high egg production strain, but this is only good for one generation. You will have to maintain the two parent lines to a high degree of productive perfection in order to continue creating the F1 hybrids which will produce for you.
In the rise of science and mass food production, it was very quickly accepted that purebred fowl are far better producers. Now we know precisely why. Before the rise of the APA, APA-sponsored egg tests, and standardized breeding for productive qualities, egg yields were much lower than what came to be the expected norm of purebred fowl.
Some people think it might be nice to start their own breed, but this is MUCH easier said than done and requires a profound understanding of genetic potential and breeding trends. Without serious study and record keeping failure is practically guaranteed.
I don't write this to be harsh, by all means. The great thing about one's homestead is that one can do what ever one pleases and, as a homestead, it doesn't really matter. As a working small-scale farm, which has a different set of demands, you would lose your shirt with a plan like that, hands down. Remember, milk and eggs in stable, predictable sums, are directly linked to the carefully managed in-breeding of pure-bred farm animals.
Best of luck!