I have a lot of welding experience, including doing it for a living at one time.
The body of a lot of axes is probably lower carbon steel than the cutting edge. If you watched the video of that old axe manufacturer from the early 1960s you can see them making the body out of one kind of steel, and the cutting edge out of another. This lets the edge be hard and lets the body be softer so it will take a shock without cracking. Japanese swords were made in a similar fashion.
Making axes this way also saves money as quality carbon tool steel is a lot more expensive than many common steels. An axe could be made out of all high-grade alloy steel, but it would have to be hardened through and then annealed everywhere except for the cutting edge and maybe the poll, a process that takes skill and would be prone to worker error as much as any process could be. With a composite metal axe the main thing that could go wrong is the weld between the two types of steel.
Someone could do a "spark" test on an axe head to see how it is made. A composite head with a welded on edge would probably produce orange-yellow sparks if the back half of it were held on a grinding wheel, and the section by the cutting edge would probably produce sparks that were more white in color. An axe head that was made completely out of one bar of steel would produce the same color sparks from front to back.
Whatever the process used to make the axe head, exotic welding methods would not be needed, but it always helps to have an educated welder. A gas-shielded mig-welder with a common good-quality wire loaded would weld any axe head made very well as long as it was not called upon to weld the cutting edge. I am sure I could "V" out the crack in this Plumb axe and lay a very nice bead on it with my gas-shielded mig welder and I could probably grind and finish it so nobody would ever know it was done.
The original point of this thread though was to suggest that the Plumb head in question had some manufacturing defect that let it crack the way it has. Some hammers for instance can be used to hammer on metal all day long as in blacksmithing and driving splitting wedges, and so the same should be possible for an axe. Accidentally hitting an axe with a hammer while installing a new handle, or using it's poll to drive splitting wedges might not be the best thing for it or the best use, but it should not crack if it shows as little use as this Plumb has had. There are a few marks on the poll from hitting something but it is nowhere near being mushroomed and has had much less use than many, many axes I have seen.
I think CityoftheSouth "hit it on the head" when he suggested that maybe Plumb made the axe a bit too hard and of one piece of steel. This could have been an effort to cut labor costs, maybe it was cheaper to make them all carbon steel and eliminate the craftsmen who shaped and welded axes together out of multiple parts by hand.
Thanks to all who gave their input!