See the website page for Legbars of Browdway http://www.legbarsofbroadway.co.uk/the%20burford%20brown/ which explains about the breeding of B. Browns and claims they are an old pure breed first developed in the 1940s and then developed for commercial production in the 1990s.
If you bred two of them together they would have the genes to produce the brown eggs, although it may be that you would not be able to buy a truebred BB cockerel from the suppliers. I am going by what happens with truebred Black Rocks, for example, where production is tightly controlled commercially and only pullets are available for purchase, and only from accredited suppliers who have raised chicks from the main hatchery to POL. This is to prevent people from breeding 'copies' of what is, in effect, a trade name for a particular product, though in the case of Rocks there are many similar 'copies,' eg Bovans Nera, resulting from a cross of the two breeds involved in this particular hybrid, ie RIR x Barred Plymouth Rock, see http://www.blackrockhens.co.uk/
The actual commercial breeding of BBs, or any other strain of bird with a particular characteristic, would be carefully controlled so that the best birds were retained in the parent flock, where there would be a female line and a separate male line so that the offspring would not become inbred.
