A very servicable round eye hawk can be made with an electric welder and a torch. Take a 2 to 2 1/2 wide bar of steel 1/4 inch thick. Heat one end and wrap it around to form the size eye you want. Then electric weld it. Make sure you get very good penetration, and add a litte extra welding to blend it all together. Then grind/sand it smooth. Cut it off to the length blade you want. Now heat up the end, and forge/flare it out to the shape blade you want. Grind/sand/smooth it all up. For occasional use, you don't even need to heat-treat it. You will have to sharpen it a little more often. If you blend that electric welded eye area well, it will be very hard to tell the difference from forge welding. Some people even heat it up and then hammer forge that electric welded area to blend it in and hide the modern weld.
A forge and all the blacksmithing tools are nice, but things can also be done with more modern tools.
Another option is taking an old single-bit axe or hatchet head, cutting off some of it, and then grinding to final shape. If you heat up the eye portion, you can then round it out into that "teardrop" shape. I've done a number of 3/4 sized axes like this - but I also cut off most of the poll and really rounded out the eye. They make a great camp axe to carry - more useful than a hawk for firewood, but easier to pack than a full-sized axe. Just another possibility.
More humble thoughts to share.
Mike Ameling