yep - take a look here
http://www.wrtcleather.com/1-ckd/hawks/hawks.html the basics are illustrated in the third row down - figure 8
While using pieces of barrel or other tubular stock was done and there are several documented originals (although overal fairly "rare"), overall plain sheet wrought iron was the most common material for the body of axes and hawks pre-1860's based on the original records as well as the archaeological records. Also rather than sawing most smiths would have split the base steel. Much easier to do when properly set up as well as virtually zero loss of what was often an expensive material.
The steel bit was usually quite narrow and thus quite often replaced - the historical record is rife with records for "re-steeling" axe and tomahawk blades.
On the other hand by the post 1860 period when large quantities of steel became more available due to the Bessemer process of steel making which increased yield and lowered prices, most axe and hawk heads were made of solid steel. Also when one sees the term "cast" steel it does not mean that the head was cast to shape, but rather it was made from a type of steel made using the Huntsman process first developed in the 1740's.
And yes Kyle is one of the best and most knowledgeable "period" smiths working today.
PS Steve looks like we were posting at the same time - isn't that one of Kyle's knives I'm doing the new sheath for and also the one I did for your sgian occles?