I have no connection to the author of this post.

I also have made many chisels for timberframing. I start with 1.5 or 2" by 3/8 or 1/2 in. thick mild steel. I thickness taper the business end and then forge weld high carbon steel (old file)to the back of the tapered end so the high carbon steel will form the cutting edge. I oil quench the working end.
For the socket, I find it quicker and easier to cut a tringular shaped piece of 1/8 sheet metal and roll it into a socket shape. I forge a short stub on the chisel and fit the socket over the stub. I then mig weld the socket to the stub/chisel.I also weld the socket seam.

I have also forge welded the socket but the mig method is faster and probably stronger. I have beaten the hell out of these chisels and the strength of the socket is a non issue.
If this is an issue for anyone, I could make a chisel for testing.I am not looking to market my chisels but am willing to improve my techniques if they are not up to standard.
The chisels pictured above, look to be very well done.
B